TITLE: Lipschitz Continuity
QUESTION [5 upvotes]: I have a quick question on how to interpret Lipschitz Continuity rather than normal continuity. What is the exact difference between the two, because I can't really find one. And is it more general than normal continuity or is it a more strict kind of continuity?

REPLY [4 votes]: Lipschitz continuity implies ordinary continuity, so it's a more strict kind. It places an upper bound on how step any secant of the function graph can be, so you may think of it as a more relaxed version of differentiability (with a bounded derivative). In fact, Rademacher's theorem states that a Lipschitz continuous function is differentiable almost everywhere (and don't worry if you don't know what “almost everywhere” means – the point is that it is mostly differentiable).
For examples of continuous functions that aren't Lipschitz continuous, look no further than $f(x)=x^{1/3}$. (Hint: Compare $f(x)-f(0)=x^{1/3}$ with $x-0=x$ for $0<x\ll 1$.)