# The domain of lsc convex functions

How to see that $int(dom(f)) \neq \emptyset$ where $f$ is a proper convex lower semi-continuous function from a Banach space to $\mathbb{R} \cup \{+\infty\}.$

A function $$f(x)=\begin{cases}+\infty&\text{for }x\ne 0\\0&\text{for }x=0\end{cases}$$ is proper convex lsc function while its domain is a singleton $\{0\}.$
• On $\Bbb R$ let $f(x_1),f(x_2)$ be finite. Then convexity itself implies that $f$ is finite on the interval $(x_1,x_2)$. Do you have any context for the question you originally asked? This could be helpful to find sufficient conditions. – szw1710 Apr 17 '17 at 19:22
• The assertion holds for any upper semicontinuous function, not necessarily convex. Namely, if $-\infty<f(x_0)<+\infty$, then $f(x)\le f(x_0)+\varepsilon$ on a neighbourhood of $x_0$. – szw1710 Apr 17 '17 at 19:39