Class of Riemann integrable functions with antiderivative

We know that continuous functions

• are Riemann integrable and
• have an antiderivative.

For each bounded function $f$ on an interval $[a, b]$, the Lebesgue integrability theorem guarantees that such an $f$ is continuous almost everywhere.

But the conditions boundedness of the function and compactness of the interval $[a, b]$ are ugly. Are there any other (hopefully huge) function classes besides continuous functions fulfilling both of these properties (having an antiderivative and being Riemann integrable)?

I posted the same question on Math.SE here a while ago, but despite its easy formulation it received no answers solving the problem.