Uniform Continuity
The ordinary
definition of continuity 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
is restricted to a certain range, for example
If a function is uniformly continuous then the same value of
can be used for allgiven the same
Uniform continuity is a stricter definition of continuity than the
criterion.
Definition A function
is uniformly continuous on
if for every
there is
such that if
with
then
If
is uniformly continuous on
we say thatis uniformly continuous.
Example:
with
Since
so
If
then
so take
and
is fixed for fixed
Example:
with
Since
so
Ifthen
so take
andis fixed for fixed