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.
 A: A classical and simple example is given by the function $x \mapsto \sin\frac{1}{x}$ extended by $0\mapsto 0$, on (say) $[-1,1]$:

*

*It is discontinuous at $0$.


*It is a derivative (= has an antiderivative) because it differs from the derivative of the differentiable function $x \mapsto x^2 \cos\frac{1}{x}$ by a continuous function (everything extended to $0$ by $0$).


*It is Riemann-integrable on $[-1,1]$ by Lebesgue's criterion, as it is bounded with a set of discontinuities $\{0\}$ having Lebesgue measure zero.
Examples with more interesting sets of discontinuity points can be constructed by summing appropriately convergent series of suitably scaled translates of this function.
It is also worth emphasizing that the fundamental theorem of calculus is valid in the following form: $F$ is differentiable and $f := F'$ is Riemann-integrable on $[a,b]$ then the Riemann integral of $f$ from $a$ to $x$ equals $F(x)-F(a)$ (note that this is with no continuity assumption on $f$, so it applies to the above example).  I don't have a reference for this at hand, but it's probably in many textbooks.
A classical reference concerning functions that are derivatives (= have an antiderivative) is Bruckner's Differentiation of Real Functions (2d ed 1994), which has various things to say about derivatives that are bounded and/or continuous a.e. (which is what you're asking about).
A: Just a modest proposal so a comment (but I am not entitled).  Every Riemann (even Lebesgue) integrable function, say on the line, can be regarded in a natural way as a distribution (the distributional derivative of its primitive which is continuous and so itself a distribution).  There is a simple and elementary (level of a freshman course in univariate analysis) theory of definite integrals for distributions which then provides an answer to your query (not necessarily the one you are looking for).  In case of interest, I would be happy to provide details and references.
