Sufficient condition(s) for $f'$ to be Riemann integrable

Question

Let $f:[0,1]\to \mathbb{R}$ be differentiable on $[0,1]$. We know that the derivative of $f$ denoted by $f'$ may not be Riemann integrable on $[0,1]$.

We know that a necessary condition for $f'$ to be Riemann integrable is boundedness of $f'$ on $[0,1]$.

My question:

1. Is boundedness of $f'$ on $[0,1]$ also a sufficient condition?

2. What are other sufficient conditions of $f'$ to be Riemann integratble on $[0,1]$?

Answer 1

The two necessary and sufficient conditions for a function $g:[a, b]\to \Bbb R$ to be Riemann integrable are

• $g$ is bounded
• $g$ is continuous almost everywhere

If $g$ happens to be a derivative $f'$, that doesn't change these requirements. Either may be broken by $f'$.

Answer 2

No, boundeness of $f'$ is not a sufficient condition. There are differentiable functions (such as Volterra's function) which are differentiable with bounded derivative whose derivative is not Riemann integrable.