Let $f: \mathbb R^n \to [0 , +\infty] $ be a lower semicontinuous and convex function. Assume $\mbox{dom} f = \{ x \in \mathbb R^n | f(x) < +\infty \}$ is a closed set. Prove or disprove that for all $x_0 \in \mbox{dom} f$, the function $f$ is continuous on $x_0$ relative to its domain ,i.e, for all sequences $x_n \to x_0$ with $f(x_n) < +\infty$ we have $f(x_n) \to f(x_0) $
Note, if $x_0 \in \mbox{relative int dom} f$ then we are done, because of convexity of $f$, thus we only need discus about case when $x_0 \in \mbox{dom}f \setminus \mbox{relative int dom} f.$
P.S This is a question motivated by the question Continuity relative to the domain of a convex function. which didn't get much attention.
Any help would be greatly appreciated .