 
Summary: SUBDIFFERENTIAL ROLLE'S AND MEAN VALUE INEQUALITY
THEOREMS
D. AZAGRA, R. DEVILLE
Abstract. In this note we give a subdifferential mean value inequality for every contin
uous G^ateaux subdifferentiable function f in a Banach space which only requires a bound
for one but not necessarily all of the subgradients of f at every point of its domain. We
also give a subdifferential approximate Rolle's theorem stating that if a subdifferentiable
function oscillates between  and on the boundary of the unit ball then there exists a
subgradient of the function at an interior point of the ball which has norm less or equal
than 2.
1. Introduction
Let X be a Banach space and U be an open convex subset of X. A function f : U  R
is said to be Fr´echet subdifferentiable at a point x U provided there exists p X
such
that
lim inf
h0
f(x + h)  f(x)  p, h
h
0,
