When do two functions have the same subdifferentials?

Question

For two functions $f$ and $g$, if $\nabla f(x) = \nabla g(x)$, $f = g + c$ for some constant $c$. Does the same hold if the gradient is replaced by the (convex) subdifferential, ie $\partial f(x) = \partial g(x)$ for all $x$ ?

And, as a stronger result, can we characterize pairs $(f, g)$ for which $\partial f(x) \cap \partial g(x) \neq \emptyset$ for all $x$ ?

Answer 1

You need some extra assumptions on $f$ and $g$. If $f,g \colon X \to \bar{\mathbb R}$ are convex and lower semicontinuous and if $X$ is a Banach space, then $\partial f = \partial g$ imply that $f$ and $g$ differ by a constant. A proof can be found in the 1970 paper "On the maximal monotonicity of subdifferential mappings" by Rockafellar, see https://doi.org/10.2140/pjm.1970.33.209.