Briefly, a function
is continuous at a point
if
is close to
for
sufficiently close to
We make this precise in the following definition.
Definition
Suppose
and
If
thenis continuous atif and only if for each
there is a
such that if
then
If
is continuous at
for every
then we sayis continuous.
Continuity and limits are not the same thing. For a function to be continuous at a point
the function must be defined at that point. A function need not be defined at a point forto have a limit at that point.
For example,
The function is not defined at 0 since
but since
so the function has a limit at x=0 but is not continuous at 0.
A more extreme, clearer, example is given by
Then
but
This is becauseonly has to tend to 0 without ever being equal to 0. In fact the value ofat a point
may have no relation to the limit of
as
This function also has a limit at
but is not continuous atsince there exists no
such that
for all
since if
and