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> This is total nonsense. The point of Lipschitz continuity is that it is more than continuity and less then differentiability. If you assert that it is differe

> This is total nonsense. The point of Lipschitz continuity is that it is more than continuity and less then differentiability. If you assert that it is differentiable the concept looses all meaning. How is Lipschitz less than differentiability? e^x is differentiable everywhere but not Lipschitz continuous everywhere, right?

constantcrying 1y ago
>How is Lipschitz less than differentiability? e^x is differentiable everywhere but not Lipschitz continuous everywhere, right? You should compare the local properties.
- dataflow 1y ago
  Nobody said local anywhere though, right? You just said the statement was nonsense where it clearly makes sense at least globally. But now that we've moved the goalposts to local: what about f(x) = x^(3/2) sin(1/x), f(0) = 0? It's differentiable yet not locally Lipschitz-continuous on (0, 1]. (Taken straight from Wikipedia)
  - constantcrying 1y ago
    >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.

constantcrying 1y ago

>How is Lipschitz less than differentiability? e^x is differentiable everywhere but not Lipschitz continuous everywhere, right? You should compare the local properties.

dataflow 1y ago
Nobody said local anywhere though, right? You just said the statement was nonsense where it clearly makes sense at least globally. But now that we've moved the goalposts to local: what about f(x) = x^(3/2) sin(1/x), f(0) = 0? It's differentiable yet not locally Lipschitz-continuous on (0, 1]. (Taken straight from Wikipedia)
- constantcrying 1y ago
  >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.