 
Summary: Mean value theorems for di erences
J. Marshall Ash
Abstract. Let M =
f(b) f(a)
b a
be the average slope of the realvalued con
tinuous function f on the closed interval [a; b]. Let 0 < p < b a: A secant
line segment connecting (c; f (c)) and (c + p; f (c + p)) of slope M for some
c 2 [a; b p] always exists when (b a) =p is an integer. But if p 2 (0; b a)
does not have the form (b a) =n for some integer n 2, then an example
is constructed for which every secant line segment lying above a subinterval
of length p does not have slope M. Applications include two counterintuitive
facts involving running certain distances at certain rates. For periodic func
tions the situation is di erent. A generalization for multivariate functions is
given.
1. One dimension
The mean value theorem says that if f (x) has a derivative at every point
x 2 (a; b) and is continuous at x = a and x = b, then there is a c 2 (a; b) such that
f0
(c) =
