Is it true that the subdifferential is empty if and only if $f$ is infinite?

Question

I am interested in conditions when the subdifferential is empty.

Claim:

If $f$ is lower semicontinuous, convex, proper, then $\partial f(x) = \emptyset$ if and only if $f(x) = +\infty$

Is this true?

I know that $f(x) = +\infty \implies \partial f(x) = \emptyset$.

But is the reverse true?

$\partial f(x) = \emptyset \implies f(x) = +\infty$?

If not, is there a general condition that says when $\partial f(x)$ is empty?

Answer 1

The function $$ f(x) = \begin{cases} -\sqrt{x} &\quad \text{if } x \geq 0, \\ \infty & \quad \text{otherwise} \end{cases} $$ is closed and convex, and $\partial f(0) = \emptyset$ despite the fact that $f(0)$ is finite.