 
Summary: PROCEEDINGS OF THE
AMERICAN MATHEMATICAL SOCIETY
Volume 101, Number 2, October 1987
MEAN VALUE THEOREMS
FOR GENERALIZEDRIEMANN DERIVATIVES
J. M. ASH AND R. L. JONES
ABSTRACT.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) hd 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 therebe 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)
h0O hd
