Mean value theorems for generalized Riemann derivatives

Abstract:

Let $x,e \geqslant 0,{u_0} < \cdots < {u_{d + e}}$ and $h > 0$ be real numbers. Let $f$ be a real valued function and let $\Delta (h;u,w)f(x){h^{ - d}}$ be a difference quotient associated with a generalized Riemann derivative. Set $I = (x + {u_0}h,x + {u_{d + e}}h)$ and let $f$ have its ordinary $(d - 1)$st derivative continuous on the closure of $I$ and its $d$th ordinary derivative ${f^{(d)}}$ existent on $I$. A necessary and sufficient condition that a difference quotient satisfy a mean value theorem (i.e., that there be a $\xi \in I$ such that the difference quotient is equal to ${f^{(d)}}(\xi ))$ is given for $d = 1$ and $d = 2$. The condition is sufficient for all $d$. It is used to show that many generalized Riemann derivatives that are "good" for numerical analysis do not satisfy this mean value theorem.