Let $C$ be a convex set, and let $f$ be a lower semi-continuous convex function. In addition, we have $f$ is bounded above on every bounded subset of $\text{ri}~C$, where $\text{ri}~C$ is the relative interior of $C$.
I know that the $f$ may not be bounded above on the whole set $C$. However, could i prove that $f$ is finite on $C$?
The definition of relative interior is \begin{equation} \text{ri}~C = \{\boldsymbol{x} \in \text{aff}~C \mid \exists \varepsilon > 0, (\boldsymbol{x}+\varepsilon B) \cap (\text{aff}~C) \subset C\}. \end{equation}
Yes, $f$ is finite on $C$. Consider a point $x$ in the Euclidean space $X$ such that $f(x) = \infty$. By lower semi-continuity, there must be some open neighbourhood $\mathcal{U}$ around $x$ such that $$y \in \mathcal{U} \implies f(y) \ge f(x) = \infty \implies f(y) = \infty.$$ If $x$ were in $C$, then the density of $\operatorname{ri} C$ in $C$ implies that $\operatorname{ri} C \cap \mathcal{U} \neq \emptyset$. This would imply the existence of a point in $\operatorname{ri} C$ where $f$ is infinite.