The pointwise characterization of derivatives of integrals

Abstract:

Let $\sigma$ denote the indefinite integral of a summable function f on ${R_n}$. We give necessary and sufficient conditions for $\sigma$ to possess a strong, or ordinary, or general derivative, equal to $f(x)$ at a point x of approximate continuity of f. Munroe [3] states that when f is a bounded function then $f(x)$ is the strong derivative of $\sigma$ at x if and only if x is a point of approximate continuity of f. We point out an error in the proof of the only if part of this result and show by example that this part of Munroe’s result is in fact false.