A function is continuous if it has no 'gaps' in it. The
criterion states this in a rigorous form which makes it possible to prove that a function is continuous or not.
The criterion states:
A function
is continuous at a point
if for
given there exists
such that
In this definition
is generally a function of
Often the function needs to be defined to an interval, so that x is restricted to a certain range, for example
The criterion is best illustrated with examples.
Example:
so if
then
Put
then 
as required for continuity.
Example:
On the interval
so
so if
then
Put
then 
as required for continuity.
Example:
On the interval
so
so ifthen
Put
then 
as required for continuity.
In this case the domain is important. The function would not be continuous if 0 were part of the domain since
is not defined for
so we can find no M such that
for all
satisfying 