# Continuity relative to the domain of a convex function.

Let $$f: \mathbb R^n \to [0 , +\infty]$$ be a lower semicontinuous, convex, and positively homogeneous degree-$$2$$ 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 it's 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 : Positively homogeneous of degree 2 means for all $$t \geq 0$$ we have $$f(tx)= t^2 f(x).$$