Continuous function

A basic concept in mathematical analysis.

Let be a real-valued function defined on a subset of the real numbers , that is, . Then is said to be continuous at a point (or, in more detail, continuous at with respect to ) if for any there exists a such that for all with the inequality

is valid. If one denotes by

and

the - and -neighbourhoods of and , respectively, then the definition above can be rephrased as follows: is called continuous at a point if for each -neighbourhood of there is a -neighbourhood of such that .

By using the concept of a limit one can say that is continuous at a point if its limit with respect to the set exists at that point and if this limit is equal to :

This is equivalent to

where , , and ; that is, to an infinitely small increment of the argument at corresponds an infinitely small increment of the function.

In terms of the limit of a sequence, the definition of continuity of a function at is: is continuous at if for every sequence of points , for which , one has

All these definitions of a function being continuous at a point are equivalent.

If is continuous at with respect to the set (or ), then is said to be continuous on the right (or left) at .

All basic elementary functions are continuous at all points of their domains of definition. An important property of continuous functions is that their class is closed under the arithmetic operations and under composition of functions. More accurately, if two real-valued functions and , , are continuous at , then so is their sum , difference and product , and when , also their quotient (which is necessarily defined in the intersection of with a certain neighbourhood of ). If, as before, is continuous at and , , is such that , so that the composite makes sense, if there is a such that and if is continuous at , then is also continuous at . Thus, in this case

that is, in this sense the operation of limit transition commutes with the operation of applying a continuous function. From these properties of continuous functions it follows that not only the basic, but also arbitrary elementary functions are continuous in their domains of definition. The property of continuity is also preserved under a uniform limit transition: If a sequence of functions converges uniformly on a set and if every is continuous at , then

is continuous at .

If a function is continuous at every point of , then is said to be continuous on the set . If and is continuous at , then the restriction of to is also continuous at . The converse is not true, in general. For example, the restriction of the Dirichlet function either to the set of rational numbers or to the set of irrational numbers is continuous, but the Dirichlet function itself is discontinuous at all points.

An important class of real-valued continuous functions of a single variable consists of those functions that are continuous on intervals. They have the following properties.

Weierstrass' first theorem: A function that is continuous on a closed interval is bounded on that interval.

Weierstrass' second theorem: A function that is continuous on a closed interval assumes on that interval a largest and a smallest value.

Cauchy's intermediate value theorem: A function that is continuous on a closed interval assumes on it any value between those at the end points.

The inverse function theorem: If a function is continuous and strictly monotone on an interval, then it has a single-valued inverse function, which is also defined on an interval and is strictly monotone and continuous on it.

Cantor's theorem on uniform continuity: A function that is continuous on a closed interval is uniformly continuous on it.

Every function that is continuous on a closed interval can be uniformly approximated on it with arbitrary accuracy by an algebraic polynomial, and every function that is continuous on and is such that can be uniformly approximated on with arbitrary accuracy by trigonometric polynomials (see Weierstrass theorem on the approximation of functions).

The concept of a continuous function can be generalized to wider forms of functions, above all, to functions of several variables. The definition above is preserved formally if one understands by a subset of an -dimensional Euclidean space , by the distance between two points and , by the -neighbourhood of in , and by

the limit of a sequence of points in . A function , , of several variables that is continuous at a point is also called continuous at this point jointly in the variables , in contrast to functions of several variables that are continuous in the variables individually. A function , , is called continuous at a point in, say, the variable if the restriction of to the set

is continuous at , that is, if the function of the single variable is continuous at . A function , , , can be continuous at in every variable , but need not be continuous at this point jointly in the variables.

The definition of a continuous function goes over directly to complex-valued functions. Only one has to interpret in the definition above as the norm of the complex number and

as the limit in the complex plane.

All these definitions are special cases of the more general concept of a continuous function with as domain of definition a certain topological space and with values in a certain topological space (see Continuous mapping).

Many properties of real-valued continuous functions of a single variable carry over to continuous mappings between topological spaces. A generalization of Weierstrass' theorem mentioned above: The continuous image of a compact topological space in a Hausdorff space is compact. A generalization of Cauchy's intermediate value theorem for a continuous function on a closed interval: A continuous image of a connected topological space in a topological space is also connected. A generalization of the theorem on the inverse function of a strictly monotone continuous function: A continuous one-to-one mapping of a compactum onto a Hausdorff space is a homeomorphism. A generalization of the theorem on the limit of a uniformly-convergent sequence of continuous functions: If is a uniformly-convergent sequence of mappings of a topological space into a metric space that are continuous (at a point ) then the limit mapping is also continuous (at ). A generalization of Weierstrass' theorem on the approximation of functions that are continuous on a closed interval is the Stone–Weierstrass theorem.

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The topics discussed above can be found in almost any introductory book on real analysis. Proofs of all statements can be found in, e.g., [a5], Chapt. 3.

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