It’s easy to prove that if $ f \in C^1([a,b], \mathbb{R})$ than $ f \in AC([a,b], \mathbb{R})$.
I was wandering what if $ f \in C^1((a,b), \mathbb{R})$. I know that it is still true, for example, if $|| f’||_{\infty, (a,b)}<\infty$, because that implies that $f$ is Lipschitz continuous. But what if $|| f’||_{\infty, (a,b)}=\infty$?
Can, someone help me, please?