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Cauchy definition of continuity (also called epsilon-delta definition): Let be a function that maps a set of real numbers to another set of real numbers. If then function is said to be continuous over at the point if for any number there exists some number such that for all with the value of satisfies . If is the domain of then is simply said to be continuous at .
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 is said to be continuous at if for any sequence such that it holds .
Cauchy’s and Heine’s definition of continuity are equivalent. The proof that if is Cauchy continuous at then is Heine continuous at is straightforward. However, to prove the converse (that Heine continuity at implies the Cauchy continuity at ) Axiom of Choice is needed.Namely, if we suppose that 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 is said to have a point of discontinuity of the first kind at if there exist both the limit from the left and the limit from the right of at and . If at least one of the one-sided limits does not exists we speak about the discontinuities of the second kind.
Function is called piecewise continuous on interval if is continuous at each point of with a finite number of exceptions, where it has discontinuities of the first kind.
Theorem (The intermediate value theorem): If the real-valued function is continuous on the closed interval and is some number between and , then there is some number such that .
Theorem (The extreme value theorem): If a function is defined on a closed and bounded set (e.g. a closed interval ) and is continuous there, then the function attains its maximum, i.e. there exists such that for all . The same is true of the minimum of .
In the - - definition of continuity of a function the value of depends on and function 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 on a closed interval is defined as .
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 with a subset of the domain of a function then function is said to be uniformly continuous at the point if for any number there exists some number such that for all satisfying we have .
A stronger property than continuity is absolute continuity. A function defined on an interval is said to be absolutely continuous if for any there exists a such that for any system of pairwise non-intersecting intervals , , for which the inequality 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 be a real valued function defined on , . Given a set of indices , let denote the value of at the point in whose th coordinate is equal to if and is equal to otherwise. For an -dimensional subinterval of , let be the alternating sum of the values of at the vertices of .
Function is said to be weakly absolutely continuous if, for any there exists such that whenever is a finite collection of disjoint rectangles in whose combined Lebesgue measure is less than . This concept was first studied by Hobson [1] , pp. 346-347.
Function is said to be strongly absolutely continuous if is weakly absolutely continuous for all non-empty (cf. [2] for the two dimensional case and [3] for general one).
[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 |
Cite this web-page as:
Štefan Porubský: Continuity of Functions.