Continuous Function

There are several commonly used methods of defining the slippery, but extremely important, concept of a continuous function (which, depending on context, may also be called a continuous map). The space of continuous functions is denoted , and corresponds to the  case of a C-k function.

A continuous function can be formally defined as a function  where the pre-image of every open set in  is open in . More concretely, a function  in a single variable  is said to be continuous at point  if

1.  is defined, so that  is in the domain of .

2.  exists for  in the domain of .

3. ,

where lim denotes a limit.

Many mathematicians prefer to define the continuity of a function via a so-called epsilon-delta definition of a limit. In this formalism, a limit  of function  as  approaches a point ,

(1)

is defined when, given any , a  can be found such that for every  in some domain  and within the neighborhood of  of radius  (except possibly  itself),

(2)

Then if  is in  and

(3)

 is said to be continuous at .

If  is differentiable at point , then it is also continuous at . If two functions  and  are continuous at , then

1.  is continuous at .

2.  is continuous at .

3.  is continuous at .

4.  is continuous at  if .

5. Providing that  is continuous at ,  is continuous at , where  denotes , the composition of the functions  and .

The notion of continuity for a function in two variables is slightly trickier, as illustrated above by the plot of the function

(4)

This function is discontinuous at the origin, but has limit 0 along the line , limit 1 along the x-axis, and limit  along the y-axis (Kaplan 1992, p. 83).

REFERENCES:

Bartle, R. G. and Sherbert, D. Introduction to Real Analysis. New York: Wiley, p. 141, 1991.

Kaplan, W. "Limits and Continuity." §2.4 in Advanced Calculus, 4th ed. Reading, MA: Addison-Wesley, pp. 82-86, 1992.