Consider a lower-semicontinuous convex function $f\colon \mathbb{R}^n \to \mathbb{R}$ with domain $C = \{x \in \mathbb{R}^d: f(x) < \infty\}$. I am interested in understanding under what conditions $f$ is continuous over $C$.

It is well known that this is true whenever $C$ is simplicial, but not otherwise (see the discussion of Theorem 10.2 in Rockafellar's convex analysis).

What if $C$ is not simplicial but $f$ very well behaved?

Is the following known: Is $f$ continuous on $C$ if $C$ is bounded and $f$ is lsc, strictly convex and essentially smooth? (essentially smooth means that $f$ is differentiable in the interior of $C$ and for every sequence $(x_n)$ in the interior of $C$, if $x_n$ converges to a point $x$ to the boundary of $C$ then $\Vert \nabla f(x_n)\Vert \to \infty$)