TITLE: What is the geometrical difference between continuity and uniform continuity?
QUESTION [45 upvotes]: Can we explain  between ordinary continuity and Uniform Continuity difference geometrically? 
What is the best way to describe the difference between these two concepts?
Where the motivation of Uniform Continuity came from? 

Thank You.

REPLY [2 votes]: Roughly speaking, geometrically in metric spaces E and F, for all ball $B_r$ in E there exists a ball $B_s$ in F such that f($B_r$) ⊂ $B_s$ . 
Two important properties of an  uniformly continuous f from E to F are : (1) the image of every Cauchy sequence in E is Cauchy in F; (2) If f applies a dense subset  of E in a COMPLET metric space F, then f admits a unique extension  to a function g which is  uniformly continuous over the whole E.This is maybe two of "motivations" you ask for.