Continuous function

There are several different definitions of (global) continuity of a function, which depend on the nature of its domain.

Using mathematical notation, there are several ways to define continuous functions in each of the three senses mentioned above.

Weierstrass and Jordan definitions (epsilon–delta) of continuous functions

Checking 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

In the same way it can be shown that the reciprocal of a continuous function

A more involved construction of continuous functions is the function composition. Given two continuous functions

Point plot of Thomae's function on the interval (0,1). The topmost point in the middle shows f(1/2) = 1/2.

For example, if a child grows from 1 m to 1.5 m between the ages of two and six years, then, at some time between two and six years of age, the child's height must have been 1.25 m.

For a Lipschitz continuous function, there is a double cone (shown in white) whose vertex can be translated along the graph, so that the graph always remains entirely outside the cone.

The translation in the language of neighborhoods of the leads to the following definition of the continuity at a point:

This definition is equivalent to the same statement with neighborhoods restricted to open neighborhoods and can be restated in several ways by using preimages rather than images.

Several equivalent definitions for a topological structure exist and thus there are several equivalent ways to define a continuous function.