The set where an approximate derivative is a derivative

Abstract:

Let $f:[0,1] \to R$ possess a finite approximate derivative $f_{\operatorname {ap}}’$ Let $E$ be the set of points $x$ where $f$ is actually differentiable. It is shown that for every $\lambda$ if $\{ x:f_{\operatorname {ap}}’(x) = \lambda \} \ne \emptyset$, then $\{ x:f_{\operatorname {ap}}’(x) = \lambda \} \cap E \ne \emptyset$. A strengthening of the mean value theorem associated with approximate derivatives is an immediate corollary.