## Illustrating Uniform Continuity:

For uniform continuity, there has to be one single
that works for a
fixed, given
.
In the picture below that is
not possible. If the
'slides' up the positive
*y*-axis, the corresponding
must get smaller and smaller. There
is no single
that will work for any possible
location of the
interval on the *y* axis.

**Not uniformly continuous**

In the example below, however, one can see that regardless of where I
place the
-interval on the *y*-axis, it is
possible to find one single small
that will work for each of those
locations of
. That is to say, there is one
that will work uniformly for all
locations of
(of course, choosing a smaller
means that I am also allowed to pick
another, smaller
- that will work again uniformly for
all -locations).

**Is uniformly continuous**