Continuous Function — from Wolfram MathWorld
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 
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).
