Is there a convex function $F$ that is not differentiable, but whose subdifferential admits a continuous selection, i.e. a continuous $g$ with $g(x) \in \partial F(x)$ for all $x$ in the domain?

In one dimension I think I can prove there is not: if $|\partial F(x)| > 1$ then it contains an open set, whose inverse image under $g$ is an open set, so $g$ is not a selection after all - I think it maps some $y \neq x$ to a point in the interior of $\partial F(x)$, which cannot be a subgradient at $y$.