Better understanding of the uniform continuity or uniform convergence

The word uniform itself explains the notions set forth.
 In my MSc 2006 , I have taken a nearly equal meaning to uniformity , that is,
"a uniformly continuous function has bounded derivative (if exists)" , actually ,this is true in it's converse sense.
   Derivative bounded means , the rate of change is in control, and function does not behave like a shock or changes abruptly.