Limit point in the subdifferential

Question

Let $f : \mathbb{R} \to \mathbb{R}$ be a convex and l.s.c. function. Take a point $x \in \mathbb{R}$ and consider the subdifferential of $f$ at $x$ denoted by $\partial f (x)$. Take any sequence $\{x_n\}_{n \ge 1} \subset \mathbb{R}$ s.t. $x_n \to x$ and an element $v \in \partial f (x)$. Is it true that there exists, up to subsequences, $\{v_n\}_{n \ge 1} \subset \mathbb{R}$ with $v_n \in \partial f(x_n)$ s.t. $v_n \to v$?

Answer 1

No, this is not always true.

Hint: Consider the function $x\mapsto |x|$.