Extended convex function - continuity

Question

I am dealing with a problem concerning a convex function defined on $\mathbb{R}^d$ and taking values on $\mathbb{R}\cup\{+\infty\}$. I would like to use in my argument that such a convex function is $\lambda^d$-almost everywhere continuous, but I do not know if this is a valid statement. Does anybody of you know a theorem or a reference which contains this claim?? Thank you in advance !

Answer 1

This statement is not true. Just consider the function $$ f(x) = \begin{cases} +\infty & \text{ if } x\ne 0\\ 0 & \text{ if } x= 0\end{cases} $$ Convex functions are continuous in the interior of their domain $dom \ f = \{x: f(x)<\infty\}$.