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 Ck function.
A continuous function can be formally defined as a function where the preimage 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 socalled epsilondelta 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 xaxis, and limit along the yaxis (Kaplan 1992, p. 83).