TITLE: Intuitive idea of the Lipschitz function
QUESTION [35 upvotes]: I'm trying to understand intuitively the notion of Lipschitz function. 
I can't understand why bounded function doesn't imply Lipschitz function.

I need a counterexample or an intuitive idea to clarify my notion of Lipschitz function.
I need help
Thanks a lot

REPLY [38 votes]: It means that your function does not explode at some point - made mathematically rigorous.
This idea can also be shown geometrically like so (picture from wikipedia):

The Lipschitz condition (or Lipschitz continuity) ensures that your function always remains entirely outside the white cone, so it cannot e.g. become infinitely steep at one point.
Concerning your picture: Think of the function $sin(1/x)$ (source W|A):

You see that it is becoming infinitely steep at the origin (and would therefore run into the white cone!) but it is still bounded.
This condition is e.g. used in the theory of differential equations where it guarantees the existence and uniqueness of solutions to certain types of differential equations.