The converse of that statement is: A continuous function must be a Lipschitz. Which I think is false.
I either wanted to use contradiction or a counterexample. But I get stuck either way.
Thank you in advance, any help is appreciated.
2026-03-30 02:05:55.1774836355
Is the converse of "a Lipschitz function must be continuous" true?
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This is false, every Lipschitz function is uniformly continuous but not every continuous function is uniformly continuous. Take $f:\mathbb R \to[0,\infty)$ where $f(x)=x^2$, this function is not uniformly continuous and so not Lipschitz, but it is continuous everywhere!