Locally Lipschitz implies continuity. Does the converse implication hold?

Intuitively, a counterexample must be a function which is very steep without having a jump or other sort of discontinuity. Consider, for example, $$g(x) = x^{1/3}$$ at $0$. Then

$$\frac{|g(x) - g(0)|}{|x - 0|} = x^{-2/3}$$

This cannot be bounded in a neighborhood of $0$.


Or $x \mapsto \sqrt{|x|}$: ${}{}{}{}$

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Real Analysis

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