TITLE: On derivatives that are not Riemann integrable
QUESTION [8 upvotes]: Let $f:\;[a,b]\to\mathbb{R}$ be differentiable on $[a,b]$. It is not a mystery that $f'$ need not be Riemann integrable. In fact even if we require $f'$ to be bounded the implication is still false. This had me wondering:
What is the weakest condition on $f$ that will guarantee a Riemann integrable derivative?

REPLY [2 votes]: A function on [a,b] is Riemann integrable if and only if it is bounded and continuous almost everywhere. (See Wikipedia.) This means that a differentiable function $f:[a,b]\rightarrow \mathbb{R}$ has a Riemann integrable derivative if and only if $f'$ is bounded and continuous almost everywhere.
We can even rephrase this to avoid mentioning $f'$ directly: a differentiable function $f:[a,b]\rightarrow \mathbb{R}$ has a Riemann integrable derivative if and only if $f$ is Lipschitz continuous (equivalent to $f'$ being bounded since we've already assumed differentiability) and "almost everywhere $C^1$".