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).
