Үздіксіз функция: математикалық анықтама және түрлері

Mathematical function with no sudden changes

In mathematics, a continuous function is a function such that a small variation of 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 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 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account at time t would be considered discontinuous since it "jumps" at each point in time when money is deposited or withdrawn.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 as follows: an infinitely small increment of the independent variable x always produces an infinitely small change of 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 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 the first published definition of uniform continuity in 1872, but based these ideas on lectures given by Peter Gustav Lejeune Dirichlet in 1854.

A neighborhood of a point c is a set that contains, at least, all points within some fixed distance of c. Intuitively, a function is continuous at a point c if the range of f over the neighborhood of c shrinks to a single point 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 there is a neighborhood in its domain such that whenever

As neighborhoods are defined in any topological space, this definition of 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.

Continuity can also be defined in terms of oscillation: a function f is continuous at a point if and only if its oscillation at that point is zero; in symbols, A benefit of this definition is that it quantifies discontinuity: the oscillation gives how much 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 (hence a set) – and gives a rapid proof of one direction of the Lebesgue integrability condition. The oscillation is equivalent to the 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 there is no that satisfies the definition, then the oscillation is at least and conversely if for every there is a desired 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 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.

The extreme value theorem states that if a function f is defined on a closed interval (or any closed and bounded set) and is continuous there, then the function attains its maximum, i. e. there exists with for all The same is true of the minimum of f. These statements are not, in general, true if the function is defined on an open interval (or any set that is not both closed and bounded), as, for example, the continuous function defined on the open interval (0,1), does not attain a maximum, being unbounded above.

Discontinuous functions may be discontinuous in a restricted way, giving rise to the concept of directional continuity (or right and left continuous functions) and semi continuity. Roughly speaking, a function is right continuous if no jump occurs when the limit point is approached from the right. Formally, f is said to be right continuous at the point c if the following holds: For any number however small, there exists some number such that for all x in the domain with the value of will satisfy

This is the same condition as continuous functions, except it is required to hold for x strictly larger than c only. Requiring it instead for all x with yields the notion of left continuous functions. A function is continuous if and only if it is both right continuous and left continuous. A function f is lower semi continuous if, roughly, any jumps that might occur only go down, but not up. That is, for any there exists some number such that for all x in the domain with the value of satisfies

The reverse condition is upper semi continuity.

Several equivalent definitions for a topological structure exist; thus, several equivalent ways exist to define a continuous function.

Continuity can also be characterized in terms of filters. A function is continuous if and only if whenever a filter on converges in to a point then the prefilter converges in to This characterization remains true if the word "filter" is replaced by "prefilter."

Symmetric to the concept of a continuous map is an open map, for which images of open sets are open. If an open map f has an inverse function, that inverse is continuous, and if a continuous map g has an inverse, that inverse is open. Given a bijective function f between two topological spaces, the inverse function need not be continuous. A bijective continuous function with a continuous inverse function is called a homeomorphism. If a continuous bijection has as its domain a compact space and its codomain is Hausdorff, then it is a homeomorphism.