Let $f$ be a continuous functions from $[0,1]$ to $\mathbb{R}$. Then, $f$ is not necessarrily lipschitz.

Question

Let $f$ be a continuous functions from $[0,1]$ to $\mathbb{R}$. Then, $f$ is not necessarily lipschitz.

Is the above statement true?

I thought since $f$ is continuous on a compact metric space, $f$ is bounded. Thus $f'$ is also bounded, which implies that $f$ is lipschitz.

Is my argument correct? If not, where did I get wrong? Help me.

Answer 1

A continuous function is not necessarily differentiable. In any case, your argument would work if the function was continuously differentiable. For a counter example, take $\sqrt x$, for instance.