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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.
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