# When is a convex function continuous on its domain?

Consider a lower-semicontinuous convex function $$fcolon mathbb{R}^n to mathbb{R}$$ 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 $$Vert nabla f(x_n)Vert to infty$$)