A proper, lower semicontinuous, convex function with no subgradient?

Question

Let $X$ be a Banach space and $f:X\to \Bbb R \cup\{\infty\}$ is a proper, lower semicontinuous and convex function.

Is it possible that $\partial f(x)=\emptyset$ for all $x\in \text{dom} f$?

If $\text{ int dom} f\ne \emptyset$ then the above situation is not possible. However, I couldn't think of a counterexample for the case $\text{ int dom} f = \emptyset$. Does anyone know if the above statement is true or false?

Answer 1

The domain of $\partial f$ is a dense subset of $\text{dom} f$, so it cannot be empty. (See e.g. Barbu-Precupanu, Convexity and optimization in Banach spaces, Corollary 2.44.)