TITLE: Intuition behind uniformly continuous functions
QUESTION [7 upvotes]: I'm trying to have an intuition behind the uniformly continuous functions. Something to show to my students. For example, before giving the formal definition and some examples of continuous functions, we can say to beginners that roughly speaking, continuous functions are functions without "holes", that's why the name "continuous". 
What about uniformly continuous functions? is there some ideas to give to the students to give them an intuition behind these functions, before to show to them the formal definition?
Thanks

REPLY [3 votes]: Here I will provide a real life heuristic in the lines of the explanation given by Michael Hardy. 
Usually when one of us move his mouse a little bit (infinitesimal displacement) it happens that the arrow on the screen moves a little bit too (infinitesimal displacement too); it is what we expects from the mouse, this is "uniformly continuous". However, sometimes one try to move a little bit the mouse (infinitesimal displacement) but the arrow remains at the same place until it suddenly jumps to some "far" place (non-infinitesimal displacement); it is what we do not expects from the mouse, this is NOT uniformly continuous).