Mean Value Theorems for Generalized Riemann Derivatives

Let x, e > 0, uo < ... u O be real numbers. Let f be a real valued function and let A (h; u, w)f (x) h-d be a difference quotient associated with a generalized Riemann derivative. Set I = (x + uoh, x + Ud+eh) and let f have its ordinary (d 1)st derivative continuous on the closure of I and its dth ordinary derivative f('I) existent on 1. A necessary and sufficient condition that a difference quotient satisfy a mean value theorem (i.e., that there be a t E I such that the difference quotient is equal to f (d)( )) 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. 1. Results. Let f be a real valued function of a real variable. The dth Riemann derivative of f is Rdaf (X) := lim = (,)( 1) f (x ?+( -d/2 + i)h) h-0O hd The first two special cases R f(x) = lim -f (x h/2) + f (x + h/2) 11-0 ~~h