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119-525: Among many other contributions, Weierstrass formalized the definition of the continuity of a function and complex analysis , proved the intermediate value theorem and the Bolzano–Weierstrass theorem , and used the latter to study the properties of continuous functions on closed bounded intervals. Weierstrass was born into a Roman Catholic family in Ostenfelde, a village near Ennigerloh , in

238-648: A δ > 0 {\displaystyle \delta >0} such that for all x ∈ D {\displaystyle x\in D} : | x − x 0 | < δ      implies      | f ( x ) − f ( x 0 ) | < ε . {\displaystyle \left|x-x_{0}\right|<\delta ~~{\text{ implies }}~~|f(x)-f(x_{0})|<\varepsilon .} More intuitively, we can say that if we want to get all

357-415: A Satz an Sich (i.e. proposition in itself) but he gives us just enough information to understand what he means by it. A proposition in itself (i) has no existence (that is: it has no position in time or place), (ii) is either true or false, independent of anyone knowing or thinking that it is true or false, and (iii) is what is 'grasped' by thinking beings. So a written sentence ('Socrates has wisdom') grasps

476-665: A copula . Instead of the more traditional copulative term 'is', Bolzano prefers 'has'. The reason for this is that 'has', unlike 'is', can connect a concrete term, such as 'Socrates', to an abstract term such as 'baldness'. "Socrates has baldness" is, according to Bolzano, preferable to "Socrates is bald" because the latter form is less basic: 'bald' is itself composed of the elements 'something', 'that', 'has' and 'baldness'. Bolzano also reduces existential propositions to this form: "Socrates exists" would simply become "Socrates has existence ( Dasein )". A major role in Bolzano's logical theory

595-526: A circle of friends and pupils who spread his thoughts about (the so-called Bolzano Circle ), but the effect of his thought on philosophy initially seemed destined to be slight. Alois Höfler (1853–1922), a former student of Franz Brentano and Alexius Meinong , who subsequently become professor of pedagogy at the University of Vienna , created the "missing link between the Vienna Circle and

714-468: A condition of his exile, Bolzano continued to develop his ideas and publish them either on his own or in obscure Eastern European journals. In 1842 he moved back to Prague, where he died in 1848. Bolzano made several original contributions to mathematics. His overall philosophical stance was that, contrary to much of the prevailing mathematics of the era, it was better not to introduce intuitive ideas such as time and motion into mathematics. To this end, he

833-527: A continuous function applies not only for real functions but also when the domain and the codomain are topological spaces and is thus the most general definition. It follows that a function is automatically continuous at every isolated point of its domain. For example, every real-valued function on the integers is continuous. One can instead require that for any sequence ( x n ) n ∈ N {\displaystyle (x_{n})_{n\in \mathbb {N} }} of points in

952-527: A corner along a given extremum and allows one to find a minimizing curve for a given integral. The lunar crater Weierstrass and the asteroid 14100 Weierstrass are named after him. Also, there is the Weierstrass Institute for Applied Analysis and Stochastics in Berlin. Continuous function In mathematics , a continuous function is a function such that a small variation of

1071-491: A correct judgment. Antonym: (a) mistake . IV. Collective meaning: Truth signifies a body or multiplicity true propositions or judgments (e.g. the biblical truth). V. Improper meaning: True signifies that some object is in reality what some denomination states it to be. (e.g. the true God). Antonyms: false, unreal, illusory . Bolzano's primary concern is with the concrete objective meaning: with concrete objective truths or truths in themselves. All truths in themselves are

1190-406: A corresponding objective idea. Schematically the whole process is like this: whenever you smell a rose, its scent causes a change in you. This change is the object of your subjective idea of that particular smell. That subjective idea corresponds to the intuition or Anschauung . According to Bolzano, all propositions are composed out of three (simple or complex) elements: a subject, a predicate and

1309-513: A domain formed by all real numbers, except some isolated points . Examples include the reciprocal function x ↦ 1 x {\textstyle x\mapsto {\frac {1}{x}}} and the tangent function x ↦ tan ⁡ x . {\displaystyle x\mapsto \tan x.} When they are continuous on their domain, one says, in some contexts, that they are continuous, although they are not continuous everywhere. In other contexts, mainly when one

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1428-637: A fruitful intellectual, and kindly personal relationship that "far transcended the usual teacher-student relationship". He mentored her for four years, and regarded her as his best student, helping to secure a doctorate for her from Heidelberg University without the need for an oral thesis defense. He was immobile for the last three years of his life, and died in Berlin from pneumonia . From 1870 until her death in 1891, Kovalevsky corresponded with Weierstrass. Upon learning of her death, he burned her letters. About 150 of his letters to her have been preserved. Professor Reinhard Bölling  [ de ] discovered

1547-399: A function f is continuous at a point x 0 {\displaystyle x_{0}} if and only if its oscillation at that point is zero; in symbols, ω f ( x 0 ) = 0. {\displaystyle \omega _{f}(x_{0})=0.} A benefit of this definition is that it quantifies discontinuity: the oscillation gives how much

1666-443: A function is continuous at a point c if the range of f over the neighborhood of c shrinks to a single point f ( c ) {\displaystyle f(c)} as the width of the neighborhood around c shrinks to zero. More precisely, a function f is continuous at a point c of its domain if, for any neighborhood N 1 ( f ( c ) ) {\displaystyle N_{1}(f(c))} there

1785-597: A great part of the Wissenschaftslehre to an explanation of these realms and their relations. Two distinctions play a prominent role in his system. First, the distinction between parts and wholes . For instance, words are parts of sentences, subjective ideas are parts of judgments, objective ideas are parts of propositions in themselves. Second, all objects divide into those that exist , which means that they are causally connected and located in time and/or space, and those that do not exist. Bolzano's original claim

1904-452: A judgment (Bolzano, Wissenschaftslehre §26). A judgment is a thought which states a true proposition. In judging (at least when the matter of the judgment is a true proposition), the idea of an object is being connected in a certain way with the idea of a characteristic (§ 23). In true judgments, the relation between the idea of the object and the idea of the characteristic is an actual/existent relation (§28). Every judgment has as its matter

2023-490: A kind of propositions in themselves. They do not exist, i.e. they are not spatiotemporally located as thought and spoken propositions are. However, certain propositions have the attribute of being a truth in itself. Being a thought proposition is not a part of the concept of a truth in itself, notwithstanding the fact that, given God's omniscience, all truths in themselves are also thought truths. The concepts 'truth in itself' and 'thought truth' are interchangeable, as they apply to

2142-403: A limit was introduced to formalize the definition of continuity. Continuity is one of the core concepts of calculus and mathematical analysis , where arguments and values of functions are real and complex numbers. The concept has been generalized to functions between metric spaces and between topological spaces . The latter are the most general continuous functions, and their definition

2261-569: A neighbourhood N ( x 0 ) {\textstyle N(x_{0})} that | f ( x ) − f ( x 0 ) | ≤ C ( | x − x 0 | )  for all  x ∈ D ∩ N ( x 0 ) {\displaystyle |f(x)-f(x_{0})|\leq C\left(\left|x-x_{0}\right|\right){\text{ for all }}x\in D\cap N(x_{0})} A function

2380-547: A popular lecturer not only in religion but also in philosophy, and he was elected Dean of the Philosophical Faculty in 1818. Bolzano alienated many faculty and church leaders with his teachings of the social waste of militarism and the needlessness of war. He urged a total reform of the educational, social and economic systems that would direct the nation's interests toward peace rather than toward armed conflict between nations. His political convictions, which he

2499-455: A proposition in itself, namely the proposition [Socrates has wisdom]. The written sentence does have existence (it has a certain location at a certain time, say it is on your computer screen at this very moment) and expresses the proposition in itself which is in the realm of in itself (i.e. an sich ). (Bolzano's use of the term an sich differs greatly from that of Kant ; for Kant's use of the term see an sich .) Every proposition in itself

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2618-421: A proposition, primarily to a proposition in itself, namely the attribute on the basis of which the proposition expresses something that in reality is as is expressed. Antonyms: falsity, falseness, falsehood . II. Concrete objective meaning: (a) Truth signifies a proposition that has the attribute truth in the abstract objective meaning. Antonym: (a) falsehood . III. Subjective meaning: (a) Truth signifies

2737-639: A proposition, which is either true or false. Every judgment exists, but not "für sich". Judgments, namely, in contrast with propositions in themselves, are dependent on subjective mental activity. Not every mental activity, though, has to be a judgment; recall that all judgments have as matter propositions, and hence all judgments need to be either true or false. Mere presentations or thoughts are examples of mental activities which do not necessarily need to be stated (behaupten), and so are not judgments (§ 34). Judgments that have as its matter true propositions can be called cognitions (§36). Cognitions are also dependent on

2856-494: A rapid proof of one direction of the Lebesgue integrability condition . The oscillation is equivalent to the ε − δ {\displaystyle \varepsilon -\delta } definition by a simple re-arrangement and by using a limit ( lim sup , lim inf ) to define oscillation: if (at a given point) for a given ε 0 {\displaystyle \varepsilon _{0}} there

2975-531: A reasonably rigorous definition of a limit as early as 1817 (and possibly even earlier) his work remained unknown to most of the mathematical community until years later, and many mathematicians had only vague definitions of limits and continuity of functions. The basic idea behind Delta-epsilon proofs is, arguably, first found in the works of Cauchy in the 1820s. Cauchy did not clearly distinguish between continuity and uniform continuity on an interval. Notably, in his 1821 Cours d'analyse, Cauchy argued that

3094-478: A rose, is this: the different aspects of the rose, like its scent and its color, cause in you a change. That change means that before and after sensing the rose, your mind is in a different state. So sensation is in fact a change in your mental state. How is this related to objects and ideas? Bolzano explains that this change, in your mind, is essentially a simple idea ( Vorstellung ), like, 'this smell' (of this particular rose). This idea represents; it has as its object

3213-457: A similar vein, Dirichlet's function , the indicator function for the set of rational numbers, D ( x ) = { 0  if  x  is irrational  ( ∈ R ∖ Q ) 1  if  x  is rational  ( ∈ Q ) {\displaystyle D(x)={\begin{cases}0&{\text{ if }}x{\text{

3332-482: A stricter relation of ' grounding ' ( Abfolge ). This is an asymmetric relation that obtains between true propositions, when one of the propositions is not only deducible from, but also explained by the other. Bolzano distinguishes five meanings the words true and truth have in common usage, all of which Bolzano takes to be unproblematic. The meanings are listed in order of properness: I. Abstract objective meaning: Truth signifies an attribute that may apply to

3451-464: A textbook, is the Theory of Science ( Wissenschaftslehre ). In the Wissenschaftslehre , Bolzano is mainly concerned with three realms: (1) The realm of language, consisting in words and sentences. (2) The realm of thought, consisting in subjective ideas and judgements. (3) The realm of logic, consisting in objective ideas (or ideas in themselves) and propositions in themselves. Bolzano devotes

3570-461: A work in four volumes that covered not only philosophy of science in the modern sense but also logic, epistemology and scientific pedagogy. The logical theory that Bolzano developed in this work has come to be acknowledged as ground-breaking. Other works are a four-volume Lehrbuch der Religionswissenschaft ( Textbook of the Science of Religion ) and the metaphysical work Athanasia , a defense of

3689-1083: Is C {\displaystyle C} -continuous for some C ∈ C . {\displaystyle C\in {\mathcal {C}}.} For example, the Lipschitz and Hölder continuous functions of exponent α below are defined by the set of control functions C L i p s c h i t z = { C : C ( δ ) = K | δ | ,   K > 0 } {\displaystyle {\mathcal {C}}_{\mathrm {Lipschitz} }=\{C:C(\delta )=K|\delta |,\ K>0\}} respectively C Hölder − α = { C : C ( δ ) = K | δ | α ,   K > 0 } . {\displaystyle {\mathcal {C}}_{{\text{Hölder}}-\alpha }=\{C:C(\delta )=K|\delta |^{\alpha },\ K>0\}.} Continuity can also be defined in terms of oscillation :

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3808-468: Is a desired δ , {\displaystyle \delta ,} the oscillation is 0. The oscillation definition can be naturally generalized to maps from a topological space to a metric space . Cauchy defined the continuity of a function in the following intuitive terms: an infinitesimal change in the independent variable corresponds to an infinitesimal change of the dependent variable (see Cours d'analyse , page 34). Non-standard analysis

3927-571: Is a neighborhood N 2 ( c ) {\displaystyle N_{2}(c)} in its domain such that f ( x ) ∈ N 1 ( f ( c ) ) {\displaystyle f(x)\in N_{1}(f(c))} whenever x ∈ N 2 ( c ) . {\displaystyle x\in N_{2}(c).} As neighborhoods are defined in any topological space , this definition of

4046-426: Is a rational number 0  if  x  is irrational . {\displaystyle f(x)={\begin{cases}1&{\text{ if }}x=0\\{\frac {1}{q}}&{\text{ if }}x={\frac {p}{q}}{\text{(in lowest terms) is a rational number}}\\0&{\text{ if }}x{\text{ is irrational}}.\end{cases}}} is continuous at all irrational numbers and discontinuous at all rational numbers. In

4165-421: Is a simple idea, it has only one object ( Einzelvorstellung ), but besides that, it is also unique (Bolzano needs this to explain sensation). Intuitions ( Anschauungen ) are objective ideas, they belong to the an sich realm, which means that they don't have existence. As said, Bolzano's argumentation for intuitions is by an explanation of sensation. What happens when you sense a real existing object, for instance

4284-401: Is a subjective idea, meaning that it is in you at a particular time. It has existence. But this subjective idea must correspond to, or has as a content, an objective idea. This is where Bolzano brings in intuitions ( Anschauungen ); they are the simple, unique and objective ideas that correspond to our subjective ideas of changes caused by sensation. So for each single possible sensation, there is

4403-475: Is a way of making this mathematically rigorous. The real line is augmented by adding infinite and infinitesimal numbers to form the hyperreal numbers . In nonstandard analysis, continuity can be defined as follows. (see microcontinuity ). In other words, an infinitesimal increment of the independent variable always produces an infinitesimal change of the dependent variable, giving a modern expression to Augustin-Louis Cauchy 's definition of continuity. Checking

4522-459: Is also continuous on D ∖ { x : g ( x ) = 0 } {\displaystyle D\setminus \{x:g(x)=0\}} . For example, the function (pictured) y ( x ) = 2 x − 1 x + 2 {\displaystyle y(x)={\frac {2x-1}{x+2}}} is defined for all real numbers x ≠ − 2 {\displaystyle x\neq -2} and

4641-439: Is at least one term that can be inserted that would make both true. A proposition Q is 'deducible' ( ableitbar ) from a proposition P, with respect to certain of their non-logical parts, if any replacement of those parts that makes P true also makes Q true. If a proposition is deducible from another with respect to all its non-logical parts, it is said to be 'logically deducible'. Besides the relation of deducibility, Bolzano also has

4760-411: Is chosen for defining them at 0 . A point where a function is discontinuous is called a discontinuity . Using mathematical notation, several ways exist to define continuous functions in the three senses mentioned above. Let f : D → R {\displaystyle f:D\to \mathbb {R} } be a function defined on a subset D {\displaystyle D} of

4879-734: Is composed out of ideas in themselves (for simplicity, we will use proposition to mean "proposition in itself" and idea to refer to an objective idea or idea in itself). Ideas are negatively defined as those parts of a proposition that are themselves not propositions. A proposition consists of at least three ideas, namely: a subject idea, a predicate idea and the copula (i.e. 'has', or another form of to have ). (Though there are propositions which contain propositions, we won't take them into consideration right now.) Bolzano identifies certain types of ideas. There are simple ideas that have no parts (as an example Bolzano uses [something]), but there are also complex ideas that consist of other ideas (Bolzano uses

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4998-443: Is continuous at a point x = x 0 {\displaystyle \displaystyle x=x_{0}} if for each x {\displaystyle x} close enough to x 0 {\displaystyle x_{0}} , the function value f ( x ) {\displaystyle f(x)} is very close to f ( x 0 ) {\displaystyle f(x_{0})} , where

5117-666: Is continuous at every such point. Thus, it is a continuous function. The question of continuity at x = − 2 {\displaystyle x=-2} does not arise since x = − 2 {\displaystyle x=-2} is not in the domain of y . {\displaystyle y.} There is no continuous function F : R → R {\displaystyle F:\mathbb {R} \to \mathbb {R} } that agrees with y ( x ) {\displaystyle y(x)} for all x ≠ − 2. {\displaystyle x\neq -2.} Since

5236-460: Is continuous everywhere apart from x = 0 {\displaystyle x=0} . Besides plausible continuities and discontinuities like above, there are also functions with a behavior, often coined pathological , for example, Thomae's function , f ( x ) = { 1  if  x = 0 1 q  if  x = p q (in lowest terms)

5355-462: Is continuous in x 0 {\displaystyle x_{0}} if it is C -continuous for some control function C . This approach leads naturally to refining the notion of continuity by restricting the set of admissible control functions. For a given set of control functions C {\displaystyle {\mathcal {C}}} a function is C {\displaystyle {\mathcal {C}}} -continuous if it

5474-680: Is continuous in D ∖ { x : f ( x ) = 0 } . {\displaystyle D\setminus \{x:f(x)=0\}.} This implies that, excluding the roots of g , {\displaystyle g,} the quotient of continuous functions q = f / g {\displaystyle q=f/g} (defined by q ( x ) = f ( x ) / g ( x ) {\displaystyle q(x)=f(x)/g(x)} for all x ∈ D {\displaystyle x\in D} , such that g ( x ) ≠ 0 {\displaystyle g(x)\neq 0} )

5593-879: Is continuous. This construction allows stating, for example, that e sin ⁡ ( ln ⁡ x ) {\displaystyle e^{\sin(\ln x)}} is continuous for all x > 0. {\displaystyle x>0.} An example of a discontinuous function is the Heaviside step function H {\displaystyle H} , defined by H ( x ) = { 1  if  x ≥ 0 0  if  x < 0 {\displaystyle H(x)={\begin{cases}1&{\text{ if }}x\geq 0\\0&{\text{ if }}x<0\end{cases}}} Pick for instance ε = 1 / 2 {\displaystyle \varepsilon =1/2} . Then there

5712-559: Is deposited or withdrawn. A form of the epsilon–delta definition of continuity was first given by Bernard Bolzano in 1817. Augustin-Louis Cauchy defined continuity of y = f ( x ) {\displaystyle y=f(x)} as follows: an infinitely small increment α {\displaystyle \alpha } of the independent variable x always produces an infinitely small change f ( x + α ) − f ( x ) {\displaystyle f(x+\alpha )-f(x)} of

5831-500: Is discontinuous at x = 0 {\displaystyle x=0} but continuous everywhere else. Yet another example: the function f ( x ) = { sin ⁡ ( x − 2 )  if  x ≠ 0 0  if  x = 0 {\displaystyle f(x)={\begin{cases}\sin \left(x^{-2}\right)&{\text{ if }}x\neq 0\\0&{\text{ if }}x=0\end{cases}}}

5950-399: Is divided into more accessible parts. Such a collection of truths is what Bolzano calls a science ( Wissenschaft ). It is important to note that not all true propositions of a science have to be known to men; hence, this is how we can make discoveries in a science. To better understand and comprehend the truths of a science, men have created textbooks ( Lehrbuch ), which of course contain only

6069-399: Is given below. Continuity of real functions is usually defined in terms of limits . A function f with variable x is continuous at the real number c , if the limit of f ( x ) , {\displaystyle f(x),} as x tends to c , is equal to f ( c ) . {\displaystyle f(c).} There are several different definitions of

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6188-650: Is interested in their behavior near the exceptional points, one says they are discontinuous. A partial function is discontinuous at a point if the point belongs to the topological closure of its domain, and either the point does not belong to the domain of the function or the function is not continuous at the point. For example, the functions x ↦ 1 x {\textstyle x\mapsto {\frac {1}{x}}} and x ↦ sin ⁡ ( 1 x ) {\textstyle x\mapsto \sin({\frac {1}{x}})} are discontinuous at 0 , and remain discontinuous whichever value

6307-484: Is irrational }}(\in \mathbb {R} \setminus \mathbb {Q} )\\1&{\text{ if }}x{\text{ is rational }}(\in \mathbb {Q} )\end{cases}}} is nowhere continuous. Bernard Bolzano Bernard Bolzano ( UK : / b ɒ l ˈ t s ɑː n oʊ / , US : / b oʊ l t ˈ s ɑː -, b oʊ l ˈ z ɑː -/ ; German: [bɔlˈtsaːno] ; Italian: [bolˈtsaːno] ; born Bernardus Placidus Johann Nepomuk Bolzano ; 5 October 1781 – 18 December 1848)

6426-415: Is no δ {\displaystyle \delta } that satisfies the ε − δ {\displaystyle \varepsilon -\delta } definition, then the oscillation is at least ε 0 , {\displaystyle \varepsilon _{0},} and conversely if for every ε {\displaystyle \varepsilon } there

6545-480: Is no δ {\displaystyle \delta } -neighborhood around x = 0 {\displaystyle x=0} , i.e. no open interval ( − δ , δ ) {\displaystyle (-\delta ,\;\delta )} with δ > 0 , {\displaystyle \delta >0,} that will force all the H ( x ) {\displaystyle H(x)} values to be within

6664-434: Is often called simply a continuous function; one also says that such a function is continuous everywhere . For example, all polynomial functions are continuous everywhere. A function is continuous on a semi-open or a closed interval; if the interval is contained in the domain of the function, the function is continuous at every interior point of the interval, and the value of the function at each endpoint that belongs to

6783-445: Is played by the notion of variations : various logical relations are defined in terms of the changes in truth value that propositions incur when their non-logical parts are replaced by others. Logically analytical propositions , for instance, are those in which all the non-logical parts can be replaced without change of truth value. Two propositions are 'compatible' ( verträglich ) with respect to one of their component parts x if there

6902-402: Is represented by an idea. An idea that has an object, represents that object. But an idea that does not have an object represents nothing. (Don't get confused here by terminology: an objectless idea is an idea without a representation.) Consider, for further explanation, an example used by Bolzano. The idea [a round square], does not have an object, because the object that ought to be represented

7021-412: Is said or asserted. "Grass", however, is only an idea ( Vorstellung ). Something is represented by it, but it does not assert anything. Bolzano's notion of proposition is fairly broad: "A rectangle is round" is a proposition — even though it is false by virtue of self- contradiction — because it is composed in an intelligible manner out of intelligible parts. Bolzano does not give a complete definition of

7140-412: Is self-contrary. A different example is the idea [nothing] which certainly does not have an object. However, the proposition [the idea of a round square has complexity] has as its subject-idea [the idea of a round square]. This subject-idea does have an object, namely the idea [a round square]. But, that idea does not have an object. Besides objectless ideas, there are ideas that have only one object, e.g.

7259-435: Is that the logical realm is populated by objects of the latter kind. Satz an Sich is a basic notion in Bolzano's Wissenschaftslehre . It is introduced at the very beginning, in section 19. Bolzano first introduces the notions of proposition (spoken or written or thought or in itself) and idea (spoken or written or thought or in itself). "The grass is green" is a proposition ( Satz ): in this connection of words, something

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7378-494: Is the basis of topology . A stronger form of continuity is uniform continuity . In order theory , especially in domain theory , a related concept of continuity is Scott continuity . As an example, the function H ( t ) denoting the height of a growing flower at time t would be considered continuous. In contrast, the function M ( t ) denoting the amount of money in a bank account at time t would be considered discontinuous since it "jumps" at each point in time when money

7497-846: The ε {\displaystyle \varepsilon } -neighborhood of H ( 0 ) {\displaystyle H(0)} , i.e. within ( 1 / 2 , 3 / 2 ) {\displaystyle (1/2,\;3/2)} . Intuitively, we can think of this type of discontinuity as a sudden jump in function values. Similarly, the signum or sign function sgn ⁡ ( x ) = {   1  if  x > 0   0  if  x = 0 − 1  if  x < 0 {\displaystyle \operatorname {sgn}(x)={\begin{cases}\;\;\ 1&{\text{ if }}x>0\\\;\;\ 0&{\text{ if }}x=0\\-1&{\text{ if }}x<0\end{cases}}}

7616-427: The f ( x 0 ) {\displaystyle f(x_{0})} neighborhood is, then f {\displaystyle f} is continuous at x 0 . {\displaystyle x_{0}.} In modern terms, this is generalized by the definition of continuity of a function with respect to a basis for the topology , here the metric topology . Weierstrass had required that

7735-441: The f ( x ) {\displaystyle f(x)} values to stay in some small neighborhood around f ( x 0 ) , {\displaystyle f\left(x_{0}\right),} we need to choose a small enough neighborhood for the x {\displaystyle x} values around x 0 . {\displaystyle x_{0}.} If we can do that no matter how small

7854-430: The product of continuous functions , p = f ⋅ g {\displaystyle p=f\cdot g} (defined by p ( x ) = f ( x ) ⋅ g ( x ) {\displaystyle p(x)=f(x)\cdot g(x)} for all x ∈ D {\displaystyle x\in D} ) is continuous in D . {\displaystyle D.} Combining

7973-409: The sum of continuous functions s = f + g {\displaystyle s=f+g} (defined by s ( x ) = f ( x ) + g ( x ) {\displaystyle s(x)=f(x)+g(x)} for all x ∈ D {\displaystyle x\in D} ) is continuous in D . {\displaystyle D.} The same holds for

8092-767: The Bauakademie to form the Technische Hochschule in Charlottenburg; now Technische Universität Berlin ). In 1864 he became professor at the Friedrich-Wilhelms-Universität Berlin, which later became the Humboldt Universität zu Berlin . In 1870, at the age of fifty-five, Weierstrass met Sofia Kovalevsky whom he tutored privately after failing to secure her admission to the university. They had

8211-577: The Province of Westphalia . Weierstrass was the son of Wilhelm Weierstrass, a government official, and Theodora Vonderforst both of whom were Catholic Rhinelanders . His interest in mathematics began while he was a gymnasium student at the Theodorianum in Paderborn . He was sent to the University of Bonn upon graduation to prepare for a government position. Because his studies were to be in

8330-548: The argument induces a small variation of the value of the function. This implies there are no abrupt changes in value, known as discontinuities . More precisely, a function is continuous if arbitrarily small changes in its value can be assured by restricting to sufficiently small changes of its argument. A discontinuous function is a function that is not continuous . Until the 19th century, mathematicians largely relied on intuitive notions of continuity and considered only continuous functions. The epsilon–delta definition of

8449-410: The fundamental theorem of algebra , which had originally been proven by Gauss from geometrical considerations. He also gave the first purely analytic proof of the intermediate value theorem (also known as Bolzano's theorem ). Today he is mostly remembered for the Bolzano–Weierstrass theorem , which Karl Weierstrass developed independently and published years after Bolzano's first proof and which

8568-555: The "close enough" restriction typically depends on the desired closeness of f ( x 0 ) {\displaystyle f(x_{0})} to f ( x ) . {\displaystyle f(x).} Using this definition, he proved the Intermediate Value Theorem. He also proved the Bolzano–Weierstrass theorem and used it to study the properties of continuous functions on closed and bounded intervals. Weierstrass also made advances in

8687-430: The (global) continuity of a function, which depend on the nature of its domain . A function is continuous on an open interval if the interval is contained in the function's domain and the function is continuous at every interval point. A function that is continuous on the interval ( − ∞ , + ∞ ) {\displaystyle (-\infty ,+\infty )} (the whole real line )

8806-478: The (pointwise) limit of (pointwise) continuous functions was itself (pointwise) continuous, a statement that is false in general. The correct statement is rather that the uniform limit of continuous functions is continuous (also, the uniform limit of uniformly continuous functions is uniformly continuous). This required the concept of uniform convergence , which was first observed by Weierstrass's advisor, Christoph Gudermann , in an 1838 paper, where Gudermann noted

8925-561: The above preservations of continuity and the continuity of constant functions and of the identity function I ( x ) = x {\displaystyle I(x)=x} on R {\displaystyle \mathbb {R} } , one arrives at the continuity of all polynomial functions on R {\displaystyle \mathbb {R} } , such as f ( x ) = x 3 + x 2 − 5 x + 3 {\displaystyle f(x)=x^{3}+x^{2}-5x+3} (pictured on

9044-399: The attention of Karl Weierstrass . To the foundations of mathematical analysis he contributed the introduction of a fully rigorous ε–δ definition of a mathematical limit . Bolzano was the first to recognize the greatest lower bound property of the real numbers. Like several others of his day, he was skeptical of the possibility of Gottfried Leibniz 's infinitesimals , that had been

9163-418: The change. Besides being simple, this change must also be unique. This is because literally you can't have the same experience twice, nor can two people, who smell the same rose at the same time, have exactly the same experience of that smell (although they will be quite alike). So each single sensation causes a single (new) unique and simple idea with a particular change as its object. Now, this idea in your mind

9282-409: The concepts figuring in this definition are subordinate to a concept of something mental or known. Bolzano proves in §§31–32 of his Wissenschaftslehre three things: There is at least one truth in itself (concrete objective meaning): B. There is more than one truth in itself: C. There are infinitely many truths in themselves: A known truth has as its parts ( Bestandteile ) a truth in itself and

9401-409: The continuity of a given function can be simplified by checking one of the above defining properties for the building blocks of the given function. It is straightforward to show that the sum of two functions, continuous on some domain, is also continuous on this domain. Given f , g : D → R , {\displaystyle f,g\colon D\to \mathbb {R} ,} then

9520-416: The definition of the limit of a function, we obtain a self-contained definition: Given a function f : D → R {\displaystyle f:D\to \mathbb {R} } as above and an element x 0 {\displaystyle x_{0}} of the domain D {\displaystyle D} , f {\displaystyle f} is said to be continuous at

9639-404: The dependent variable y (see e.g. Cours d'Analyse , p. 34). Cauchy defined infinitely small quantities in terms of variable quantities, and his definition of continuity closely parallels the infinitesimal definition used today (see microcontinuity ). The formal definition and the distinction between pointwise continuity and uniform continuity were first given by Bolzano in the 1830s, but

9758-940: The domain of f {\displaystyle f} with x 0 − δ < x < x 0 + δ , {\displaystyle x_{0}-\delta <x<x_{0}+\delta ,} the value of f ( x ) {\displaystyle f(x)} satisfies f ( x 0 ) − ε < f ( x ) < f ( x 0 ) + ε . {\displaystyle f\left(x_{0}\right)-\varepsilon <f(x)<f(x_{0})+\varepsilon .} Alternatively written, continuity of f : D → R {\displaystyle f:D\to \mathbb {R} } at x 0 ∈ D {\displaystyle x_{0}\in D} means that for every ε > 0 , {\displaystyle \varepsilon >0,} there exists

9877-493: The domain of f {\displaystyle f} ,     | x − x 0 | < δ ⇒ | f ( x ) − f ( x 0 ) | < ε . {\displaystyle \displaystyle \ |x-x_{0}|<\delta \Rightarrow |f(x)-f(x_{0})|<\varepsilon .} In simple English, f ( x ) {\displaystyle \displaystyle f(x)}

9996-409: The domain of f , exists and is equal to f ( c ) . {\displaystyle f(c).} In mathematical notation, this is written as lim x → c f ( x ) = f ( c ) . {\displaystyle \lim _{x\to c}{f(x)}=f(c).} In detail this means three conditions: first, f has to be defined at c (guaranteed by

10115-820: The domain which converges to c , the corresponding sequence ( f ( x n ) ) n ∈ N {\displaystyle \left(f(x_{n})\right)_{n\in \mathbb {N} }} converges to f ( c ) . {\displaystyle f(c).} In mathematical notation, ∀ ( x n ) n ∈ N ⊂ D : lim n → ∞ x n = c ⇒ lim n → ∞ f ( x n ) = f ( c ) . {\displaystyle \forall (x_{n})_{n\in \mathbb {N} }\subset D:\lim _{n\to \infty }x_{n}=c\Rightarrow \lim _{n\to \infty }f(x_{n})=f(c)\,.} Explicitly including

10234-466: The draft of the letter she wrote to Weierstrass when she arrived in Stockholm in 1883 upon her appointment as Privatdocent at Stockholm University . Weierstrass was interested in the soundness of calculus, and at the time there were somewhat ambiguous definitions of the foundations of calculus so that important theorems could not be proven with sufficient rigour. Although Bolzano had developed

10353-403: The earliest putative foundation for differential calculus . Bolzano's notion of a limit was similar to the modern one: that a limit, rather than being a relation among infinitesimals, must instead be cast in terms of how the dependent variable approaches a definite quantity as the independent variable approaches some other definite quantity. Bolzano also gave the first purely analytic proof of

10472-479: The example of [nothing], which consists of the ideas [not] and [something]). Complex ideas can have the same content (i.e. the same parts) without being the same — because their components are differently connected. The idea [A black pen with blue ink] is different from the idea [A blue pen with black ink] though the parts of both ideas are the same. It is important to understand that an idea does not need to have an object. Bolzano uses object to denote something that

10591-545: The field of calculus of variations . Using the apparatus of analysis that he helped to develop, Weierstrass was able to give a complete reformulation of the theory that paved the way for the modern study of the calculus of variations. Among several axioms, Weierstrass established a necessary condition for the existence of strong extrema of variational problems. He also helped devise the Weierstrass–Erdmann condition , which gives sufficient conditions for an extremal to have

10710-472: The fields of law, economics, and finance, he was immediately in conflict with his hopes to study mathematics. He resolved the conflict by paying little heed to his planned course of study but continuing private study in mathematics. The outcome was that he left the university without a degree. He then studied mathematics at the Münster Academy (which was even then famous for mathematics) and his father

10829-511: The first published definition of uniform continuity in 1872, but based these ideas on lectures given by Peter Gustav Lejeune Dirichlet in 1854. A real function that is a function from real numbers to real numbers can be represented by a graph in the Cartesian plane ; such a function is continuous if, roughly speaking, the graph is a single unbroken curve whose domain is the entire real line. A more mathematically rigorous definition

10948-439: The function sine is continuous on all reals, the sinc function G ( x ) = sin ⁡ ( x ) / x , {\displaystyle G(x)=\sin(x)/x,} is defined and continuous for all real x ≠ 0. {\displaystyle x\neq 0.} However, unlike the previous example, G can be extended to a continuous function on all real numbers, by defining

11067-418: The function is discontinuous at a point. This definition is helpful in descriptive set theory to study the set of discontinuities and continuous points – the continuous points are the intersection of the sets where the oscillation is less than ε {\displaystyle \varepsilon } (hence a G δ {\displaystyle G_{\delta }} set ) – and gives

11186-476: The fundamental truths that may or may not appear to be obvious to our intuitions. Bolzano begins his work by explaining what he means by theory of science , and the relation between our knowledge, truths and sciences. Human knowledge, he states, is made of all truths (or true propositions) that men know or have known. However, this is a very small fraction of all the truths that exist, although still too much for one human being to comprehend. Therefore, our knowledge

11305-474: The idea [the first man on the moon] represents only one object. Bolzano calls these ideas 'singular ideas'. Obviously there are also ideas that have many objects (e.g. [the citizens of Amsterdam]) and even infinitely many objects (e.g. [a prime number]). Bolzano has a complex theory of how we are able to sense things. He explains sensation by means of the term intuition, in German called Anschauung . An intuition

11424-609: The immortality of the soul. Bolzano also did valuable work in mathematics, which remained virtually unknown until Otto Stolz rediscovered many of his lost journal articles and republished them in 1881. In his 1837 Wissenschaftslehre Bolzano attempted to provide logical foundations for all sciences, building on abstractions like part-relation, abstract objects , attributes, sentence-shapes, ideas and propositions in themselves, sums and sets , collections, substances, adherences, subjective ideas, judgments, and sentence-occurrences. These attempts were an extension of his earlier thoughts in

11543-422: The interval x 0 − δ < x < x 0 + δ {\displaystyle x_{0}-\delta <x<x_{0}+\delta } be entirely within the domain D {\displaystyle D} , but Jordan removed that restriction. In proofs and numerical analysis, we often need to know how fast limits are converging, or in other words, control of

11662-482: The interval is the limit of the values of the function when the variable tends to the endpoint from the interior of the interval. For example, the function f ( x ) = x {\displaystyle f(x)={\sqrt {x}}} is continuous on its whole domain, which is the closed interval [ 0 , + ∞ ) . {\displaystyle [0,+\infty ).} Many commonly encountered functions are partial functions that have

11781-769: The phenomenon but did not define it or elaborate on it. Weierstrass saw the importance of the concept, and both formalized it and applied it widely throughout the foundations of calculus. The formal definition of continuity of a function, as formulated by Weierstrass, is as follows: f ( x ) {\displaystyle \displaystyle f(x)} is continuous at x = x 0 {\displaystyle \displaystyle x=x_{0}} if ∀   ε > 0   ∃   δ > 0 {\displaystyle \displaystyle \forall \ \varepsilon >0\ \exists \ \delta >0} such that for every x {\displaystyle x} in

11900-434: The philosophy of mathematics, for example his 1810 Beiträge where he emphasized the distinction between the objective relationship between logical consequences and our subjective recognition of these connections. For Bolzano, it was not enough that we merely have confirmation of natural or mathematical truths, but rather it was the proper role of the sciences (both pure and applied) to seek out justification in terms of

12019-415: The point x 0 {\displaystyle x_{0}} when the following holds: For any positive real number ε > 0 , {\displaystyle \varepsilon >0,} however small, there exists some positive real number δ > 0 {\displaystyle \delta >0} such that for all x {\displaystyle x} in

12138-467: The remainder. We can formalize this to a definition of continuity. A function C : [ 0 , ∞ ) → [ 0 , ∞ ] {\displaystyle C:[0,\infty )\to [0,\infty ]} is called a control function if A function f : D → R {\displaystyle f:D\to R} is C -continuous at x 0 {\displaystyle x_{0}} if there exists such

12257-428: The requirement that c is in the domain of f ). Second, the limit of that equation has to exist. Third, the value of this limit must equal f ( c ) . {\displaystyle f(c).} (Here, we have assumed that the domain of f does not have any isolated points .) A neighborhood of a point c is a set that contains, at least, all points within some fixed distance of c . Intuitively,

12376-466: The right). In the same way, it can be shown that the reciprocal of a continuous function r = 1 / f {\displaystyle r=1/f} (defined by r ( x ) = 1 / f ( x ) {\displaystyle r(x)=1/f(x)} for all x ∈ D {\displaystyle x\in D} such that f ( x ) ≠ 0 {\displaystyle f(x)\neq 0} )

12495-442: The same objects, but they are not identical. Bolzano offers as the correct definition of (abstract objective) truth: a proposition is true if it expresses something that applies to its object. The correct definition of a (concrete objective) truth must thus be: a truth is a proposition that expresses something that applies to its object. This definition applies to truths in themselves, rather than to thought or known truths, as none of

12614-473: The set R {\displaystyle \mathbb {R} } of real numbers. This subset D {\displaystyle D} is the domain of f . Some possible choices include In the case of the domain D {\displaystyle D} being defined as an open interval, a {\displaystyle a} and b {\displaystyle b} do not belong to D {\displaystyle D} , and

12733-1107: The sinc-function becomes a continuous function on all real numbers. The term removable singularity is used in such cases when (re)defining values of a function to coincide with the appropriate limits make a function continuous at specific points. A more involved construction of continuous functions is the function composition . Given two continuous functions g : D g ⊆ R → R g ⊆ R  and  f : D f ⊆ R → R f ⊆ D g , {\displaystyle g:D_{g}\subseteq \mathbb {R} \to R_{g}\subseteq \mathbb {R} \quad {\text{ and }}\quad f:D_{f}\subseteq \mathbb {R} \to R_{f}\subseteq D_{g},} their composition, denoted as c = g ∘ f : D f → R , {\displaystyle c=g\circ f:D_{f}\to \mathbb {R} ,} and defined by c ( x ) = g ( f ( x ) ) , {\displaystyle c(x)=g(f(x)),}

12852-441: The subject, and so, opposed to truths in themselves, cognitions do permit degrees; a proposition can be more or less known, but it cannot be more or less true. Every cognition implies necessarily a judgment, but not every judgment is necessarily cognition, because there are also judgments that are not true. Bolzano maintains that there are no such things as false cognitions, only false judgments (§34). Bolzano came to be surrounded by

12971-399: The true propositions of the science known to men. But how to know where to divide our knowledge, that is, which truths belong together? Bolzano explains that we will ultimately know this through some reflection, but that the resulting rules of how to divide our knowledge into sciences will be a science in itself. This science, that tells us which truths belong together and should be explained in

13090-418: The value G ( 0 ) {\displaystyle G(0)} to be 1, which is the limit of G ( x ) , {\displaystyle G(x),} when x approaches 0, i.e., G ( 0 ) = lim x → 0 sin ⁡ x x = 1. {\displaystyle G(0)=\lim _{x\to 0}{\frac {\sin x}{x}}=1.} Thus, by setting

13209-413: The values of f ( a ) {\displaystyle f(a)} and f ( b ) {\displaystyle f(b)} do not matter for continuity on D {\displaystyle D} . The function f is continuous at some point c of its domain if the limit of f ( x ) , {\displaystyle f(x),} as x approaches c through

13328-490: The widow of his friend Carl Wilhelm Borchardt . After 1850 Weierstrass suffered from a long period of illness, but was able to publish mathematical articles that brought him fame and distinction. The University of Königsberg conferred an honorary doctor's degree on him on 31 March 1854. In 1856 he took a chair at the Gewerbeinstitut in Berlin (an institute to educate technical workers which would later merge with

13447-456: The work wasn't published until the 1930s. Like Bolzano, Karl Weierstrass denied continuity of a function at a point c unless it was defined at and on both sides of c , but Édouard Goursat allowed the function to be defined only at and on one side of c , and Camille Jordan allowed it even if the function was defined only at c . All three of those nonequivalent definitions of pointwise continuity are still in use. Eduard Heine provided

13566-602: Was a Bohemian mathematician , logician , philosopher , theologian and Catholic priest of Italian extraction, also known for his liberal views. Bolzano wrote in German , his native language. For the most part, his work came to prominence posthumously. Bolzano was the son of two pious Catholics . His father, Bernard Pompeius Bolzano, was an Italian who had moved to Prague , where he married Maria Cecilia Maurer who came from Prague's German-speaking family Maurer. Only two of their twelve children lived to adulthood. When he

13685-818: Was able to obtain a place for him in a teacher training school in Münster . Later he was certified as a teacher in that city. During this period of study, Weierstrass attended the lectures of Christoph Gudermann and became interested in elliptic functions . In 1843 he taught in Deutsch Krone in West Prussia and from 1848 he taught at the Lyceum Hosianum in Braunsberg . Besides mathematics he also taught physics, botany, and gymnastics. Weierstrass may have had an illegitimate child named Franz with

13804-489: Was inclined to share with others with some frequency, eventually proved to be too liberal for the Austrian authorities. On December 24, 1819, he was removed from his professorship (upon his refusal to recant his beliefs) and was exiled to the countryside and then devoted his energies to his writings on social, religious, philosophical, and mathematical matters. Although forbidden to publish in mainstream journals as

13923-564: Was initially called the Weierstrass theorem until Bolzano's earlier work was rediscovered. Bolzano's posthumously published work Paradoxien des Unendlichen (The Paradoxes of the Infinite) (1851) was greatly admired by many of the eminent logicians who came after him, including Charles Sanders Peirce , Georg Cantor , and Richard Dedekind . Bolzano's main claim to fame, however, is his 1837 Wissenschaftslehre ( Theory of Science ),

14042-438: Was one of the earliest mathematicians to begin instilling rigor into mathematical analysis with his three chief mathematical works Beyträge zu einer begründeteren Darstellung der Mathematik (1810), Der binomische Lehrsatz (1816) and Rein analytischer Beweis (1817). These works presented "...a sample of a new way of developing analysis", whose ultimate goal would not be realized until some fifty years later when they came to

14161-590: Was ten years old, Bolzano entered the Gymnasium of the Piarists in Prague, which he attended from 1791 to 1796. Bolzano entered the University of Prague in 1796 and studied mathematics , philosophy and physics . Starting in 1800, he also began studying theology , becoming a Catholic priest in 1804. He was appointed to the new chair of philosophy of religion at Prague University in 1805. He proved to be

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