Singularities of Fitzpatrick and Convex Functions

In a pseudo-Euclidean space with scalar product $S(\cdot, \cdot)$, we show that the singularities of projections on $S$-monotone sets and of the associated Fitzpatrick functions are covered by countable $c-c$ surfaces having positive normal vectors with respect to the $S$-product. By L. Zajíček [On the differentiation of convex functions in finite and infinite dimensional spaces, Czechoslovak Math. J. 29/104 (1979) 340–348], the singularities of a convex function $f$ can be covered by a countable collection of $c-c$ surfaces. We show that the normal vectors to these surfaces are restricted to the cone generated by $F-F$, where $F:= {\rm cl}\,{\rm range}\,\nabla f$, the closure of the range of the gradient of $f$.