fa.functional analysis - Local supporting points of Lipschitz functions

Let X be a separable reflexive Banach space and f:Xtomathbb{R} be a
Lipschitz function. Say that a point x in X is a local supporting point
of f if there exist x^* in X^* and an open neighborhood U of x
such that either x^* (y-x)leq f(y)-f(x) for all y in U or
x^* (y-x)geq f(y)-f(x) for all y in U.

Question: is true that the set of local supporting points of f is dense in X?

This question is obviously related to differentiability; it might be difficult.

I would be very much interested to know whether it has been asked by others.