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

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?

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.)