>Nobody said local anywhere though, right? I was speaking in very general terms about the subject, I was not making mathematically correct statements. Your are

>Nobody said local anywhere though, right? I was speaking in very general terms about the subject, I was not making mathematically correct statements. Your are completely correct about the counter example. You actually need some stronger assumptions, namely that f is C^1, meaning it has a continous derivative. In that case any function f defined on a compact set K has a continuous derivative f' which assumes its maximum on K, giving Lipschitz continuity. Every function in C^1 on K is Lipschitz, yet not every Lipschitz function is C^1 or even differentiable. To be more specific and to give the reason Lipschitz functions are important, look at the following statement from Wikipedia. "A Lipschitz function g : R → R is absolutely continuous and therefore is differentiable almost everywhere, that is, differentiable at every point outside a set of Lebesgue measure zero." Given this I hope that you can understand what I said about Lipschitz functions being a weaker version of differentiabilty.