The ordinarydefinition of continuity states:
A functionis 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 ofcan be used for all
given the same
Uniform continuity is a stricter definition of continuity than the
criterion.
Definition A functionis uniformly continuous on
if for every
there is
such that if
with
then
If
is uniformly continuous on
we say that
is uniformly continuous.
Example:with
Sinceso
Ifthen
so take
and
is fixed for fixed
Example:with
Sinceso
Ifthen
so take
and
is fixed for fixed