On functions with summable approximate Peano derivative

Abstract:

Let $n$ be a positive integer and $F$ a function defined on a closed interval $I$. For $x$ in $I$, let the $n$th approximate Peano derivative of $F$ at $x$, if it exists, be denoted as ${F_{(n)}}(x)$. For $n = 1$, the existence of ${F_{(n - 1)}}(x)$ will simply mean that the function ${F_{(0)}}( \equiv F)$ is approximately continuous at $x$. Then the following theorem is proved, noting that the phrase “for nearly all $x$ in $I$” means “for all $x$ in $I$ except perhaps for those points in a countable subset of $I$", Theorem ${{\mathbf {A}}_n}$. Let ${F_{(n - 1)}}(x)$ exist finitely for all $x$ in $I$. If ${F_{(n)}}(x)$ exists finitely for nearly all $x$ in $I$ and is summable on $I$, then ${F_{(n - 1)}}$ is absolutely continuous in $I$.