Can a non-constant lower-semi-continuous convex function attain the $-\infty$ value This paper that I am studying asserts in the second paragraph that the only lower-semi-continuous convex function attaining $-\infty$ is the constant $-\infty$ function. However, it seems that the function defined by:
\begin{align}
f(x)=\begin{cases}
-\infty, &x\leq 0\\
\infty, &x>0
\end{cases}
\end{align}
is both convex and lower-semi-continuous. My reasoning is that the epigraph of $f$ is a closed half-plane, which I know to be convex. For lower-semi-continuity, it seems that the only point of concern is $x=0$. But $f(0)=-\infty$, and so it is clear that for all $\epsilon>0$, we have $f(x) \geq f(0)-\epsilon = -\infty$ for any $x\in \mathbb{R}$.
Is there something I'm missing here? Is there something else we assume in the definition of convexity? Am I misinterpreting the statement in the paper?

You're correct, and the paper is wrong.
