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Continuity of Functions

Cauchy definition of continuity (also called epsilon-delta definition): Let typeset structure be a function that maps a set typeset structure of real numbers to another set typeset structure of real numbers. If typeset structure then function typeset structure is said to be continuous over typeset structure at the point typeset structure if for any number typeset structure there exists some number typeset structure such that for all typeset structure with typeset structure the value of typeset structure satisfies typeset structure. If typeset structure is the domain of typeset structure then typeset structure is simply said to be continuous at typeset structure.

We say that a function is continuous on some subset of its domain if it is continuous at every point of that subset. It is call everywhere continuous, or simply continuous, if it is continuous at every point of its domain.

Heine definition of continuity: A function typeset structure is said to be continuous at typeset structure if for any sequence typeset structure such that typeset structure it holds typeset structure.

Cauchy’s and Heine’s definition of continuity are equivalent. The proof that if typeset structure  is Cauchy continuous at typeset structure then typeset structure  is Heine continuous at typeset structure  is straightforward. However, to prove the converse (that Heine continuity at typeset structure implies the Cauchy continuity at typeset structure)  Axiom of Choice is needed.Namely, if we suppose that typeset structure is not Cauchy continuous then using the Axiom of Choice we get a sequence whose existence contradicts Heine continuity.

In the case of global continuity of real functions it was proved by Wacław Sierpiński that the axiom of choice is not actually needed

Function typeset structure is said to have a point of discontinuity of the first kind at typeset structure if there exist both the limit typeset structurefrom the left and the limit typeset structure from the right of typeset structure at typeset structure and typeset structure. If at least one of the one-sided limits does not exists we speak about the discontinuities of the second kind.

Function typeset structure is called piecewise continuous on interval typeset structure if typeset structure is continuous at each point of typeset structure with a finite number of exceptions, where it has discontinuities of the first kind.

Theorem (The intermediate value theorem): If the real-valued function typeset structure is continuous on the closed interval typeset structure and typeset structure is some number between typeset structure and typeset structure, then there is some number typeset structure such that typeset structure.

Theorem (The extreme value theorem): If a function typeset structure is defined on a closed and bounded set (e.g. a closed interval typeset structure) and is continuous there, then the function attains its maximum, i.e. there exists typeset structure such that typeset structurefor all typeset structure. The same is true of the minimum of typeset structure.

In the typeset structure-typeset structure - definition of continuity of a function typeset structure the value of typeset structure depends on typeset structure and function typeset structure as well. This kind of dependence can be described by a function, so called, (global) modulus of continuity formally introduced by H.Lebesgue in 1910: The modulus of continuity of a continuous function typeset structure on a closed interval is defined as typeset structure.

In more general setting of topological spaces, the concept analogous to Heine definition of continuity is called sequential continuity, and the sequential continuity is not equivalent to the analogue of Cauchy continuity.

If typeset structure with typeset structure a subset of the domain of a function typeset structure then function typeset structure is said to be uniformly continuous at the point typeset structure if for any number typeset structure there exists some number typeset structure such that for all typeset structure  satisfying  typeset structure we have typeset structure.

A stronger property than continuity is absolute continuity. A function typeset structure defined on an interval typeset structure is said to be absolutely continuous if for any typeset structure there exists a typeset structure such that for any system of pairwise non-intersecting intervals typeset structure, typeset structure, for which typeset structure the inequality typeset structure holds.

If, in the above definition, the requirement that the pairwise intersections of intervals  are empty is dropped, then the function is said the so called Lipschitz condition.

In the case of multivariate functions there is a parallelism between notions connected with absolute continuity and that of  bounded Vitali and Hardy-Krause variation .

Let typeset structure be a real valued function defined on typeset structure, typeset structure.  Given a set of indices typeset structure, let typeset structure denote the value of typeset structure at the point in typeset structure whose typeset structureth coordinate is equal to typeset structure if typeset structure and is equal to typeset structure otherwise. For an typeset structure-dimensional subinterval typeset structure of typeset structure, let typeset structure be the alternating sum of the values of typeset structure at the vertices of typeset structure.

Function typeset structure is said to be weakly absolutely continuous if, for any typeset structurethere exists typeset structure such that typeset structure whenever typeset structure is a finite collection of disjoint rectangles in typeset structure whose combined Lebesgue measure is less than typeset structure.  This concept was first studied by Hobson  [1] , pp. 346-347.

Function typeset structure is said to be strongly absolutely continuous if typeset structure is weakly absolutely continuous for all non-empty typeset structure (cf. [2] for the two dimensional case and  [3] for general one).   

References

[1]  Hobson, E. W. (1927). The Theory of Functions of a Real Variable and the Theory of Fourier’s Series, Vol. 1 (3rd ed.). Cambridge University Press (Republished by Harren Press 1950).

[2]  Berkson, E., & Gillespie, T. A. (1984). Absolutely continuous functions of two variables and well-bounded operators. J. London Math. Soc. (2), 30, 305-321.

[3]  Beare, B. K. (2007). Copulas and temporal dependence. Job Market Paper http://www.nuffield.ox.ac.uk/General/Seminars/Papers/584.pdf

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Štefan Porubský: Continuity of Functions.

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