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