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.
1. is defined, so that is in the domain of .
2. exists for in the domain of .
where lim denotes a limit.
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),
Then if is in and
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