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I have a question about the fine structure of convex functions. Convex functions behave very regular in the interior of their domain of definition (e.g. they are locally Lipschitz continuous there) but otherwise some weird things can happen. My question concerns convex functions that possess at most one subgradient at each point. Let's fix notation:

Let $X$ be a Banach space and $J:X\to\mathbb{R}\cup\{\infty\}$ be a convex, extended valued function. Denote by $\newcommand{\dom}{\mathrm{dom}}\dom J = \{x\ :\ J(x)<\infty\}$ and assume that the subdifferential $\partial J$ of $J$ is at most single valued and denote its unique element by $\nabla J(x)$ (if it exists). Moreover, denote the Gâteaux directional derivative at $x$ in direction $h$ by $DJ(x;h)$. My question is:

Does $x,y\in \dom J$ imply that $$\langle \nabla J(x),y-x\rangle = DJ(x;y-x)\ ?$$

Some background: I would like to state that in the above framework for some non-strictly convex $J$ there exist $x,y\in\dom J$ such that $\langle \nabla J(x),y-x\rangle = J(y)-J(x)$. It clear that one gets $x$ and $y$ such that for $\lambda\in]0,1]$ it holds that $$ \frac{J(\lambda y + (1-\lambda)x) - J(x)}{\lambda}=J(y) - J(x) $$ which implies $DJ(x,y-x) = J(y) - J(x)$.

However, there exists a pathological convex function such that its subdifferential at some point is single values although it is not Gâteaux differentiable there (Example 4.2.6 in Borwein and Vanderwerffs "Convex Functions: Constructions, Characterizations and Counterexamples", see here). However, I assume a wee bit more, namely that the subdifferential is at most single valued everywhere (but probably this already rules out some pathological things…).

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What about $f(x) = -1(1-|x|^2)^{1/2}$ on $|x| \le 1$ and $f(x)=+\infty$ elsewhere? This function is in fact differentiable at $x$ for all $|x| < 1$, but $\partial f(x) = \emptyset$ for all $|x| \ge 1$---the badness comes from $1 \in \dom f$--- does this break your setup? –  Suvrit Dec 20 '13 at 17:26
    
Well, I am interested in the case where $\partial f(x)$ is a singleton for all $x$ in $\mathrm{dom} f$. In your example, the subdifferential is empty precisely where the directional derivative does not exist. –  Dirk Dec 20 '13 at 18:02
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