Let $f: X \to [ -\infty, \infty]$ be some function,


Can someone provide a non-trivial example where the subdifferential evaluated at a point $x$,
$$\partial f(x)$$ is "unbounded"? (trivial examples included the improper functions)


A rough definition of an unbounded subgradient is that there exists some sequence $v\_n \in \partial f(x)$ such that $\|v\_n\| \to \infty$. 



What is a non-trivial example of an unbounded subdifferential?
A simple example is the convex function
$$
f(x) = \begin{cases} \infty, & \text{if}\ x<0\\
x & \text{if}\ x\geq 0.
\end{cases}
$$
It holds that $\partial f(0) = ]-\infty,1]$.


There are no examples without the value $\infty$: If $f$ is convex and bounded in a neighborhood at some point, then $f$ is locally Lipschitz at that point and hence, the subgradient at that point is bounded.


If you do not assume convexity, you can get unbounded subgradients even for bounded functions as a comment to the question shows.


