Consider a sequence of continuous convex functions, $f_n,f:A \rightarrow \mathbb{R}$, where $A \subseteq \mathbb{R}^n$, compact. The sequence $f_n$ converges uniformly to $f$. Does $\nabla f_n$ converge to $\nabla f$ pointwise, wherever $\nabla f$ exists?
I am confused because this paper and this paper seem to suggest that the answer is yes (See footnote), but this answer (See the second answer) seems to provide a counterexample.
What am I missing?
Edit: The following theorem in Rockafellar's Convex Analysis also seems to imply pointwise convergence of derivatives wherever they exist (Note the "Moreover" part).
THEOREM 24.5 (Rockafellar, 1997, p. 233). Let $f$ be a convex function on $R^n$, and let $C$ be an open convex set on which $f$ is finite. Let $f_1, f_2, \ldots$, be a sequence of convex functions finite on $C$ and converging pointwise to f on $C$. Let $x \in C$, and let $x_1, x_2, \ldots$, be a sequence of points in $C$ converging to $x$. Then, for any $y \in R^n$ and any sequence $y_1, y_2, \ldots$, converging to $y$, one has
$$
\limsup _{i \rightarrow \infty} f_i^{\prime}\left(x_i ; y_i\right) \leq f^{\prime}(x ; y) .
$$
Moreover, given any $\varepsilon>0$, there exists an index $i_0$ such that
$$
\partial f_i\left(x_i\right) \subset \partial f(x)+\varepsilon B, \quad \forall i \geq i_0,
$$
where $B$ is the Euclidean unit ball of $R^n$.
Footnote: The first paper mentions this right at the first paragraph. In the second paper, towards the beginning of the second page, the author says,"Roughly speaking, a sequence of convex functions converges if and only if the sequence of their "derivatives" converges, too!" This is right after he gives the precise statement of this result, starting at the bottom of the first page.
Best Answer
Convergence of subdifferentials is usually meant in the sense of graphical convergence, which is not the same thing as pointwise convergence. The convergence mentioned by the theorem of Attouch refers to graphical convergence, whereas (I believe) the counterexample demonstrates that pointwise convergence may fail. The distinction is a bit technical, so I'll just point to a really great reference with formal definitions, motivating examples, comparisons, and theoretical guarantees (e.g., when one type of convergence can imply another).
Variational Analysis, by Rockafellar and Wets,
Chapter 5.E (page 166): these notions of convergence and their differences
Theorem 12.35: The convergence result by Attouch.
p. 239 - discussion on pointwise convergence
To quote the book,