# Convex non-increasing, but not lower semicontinuous function?

I am trying to think of an example of a function $f:\unicode{x211D}^n \rightarrow \unicode{x211D}$ that is convex and non-increasing, but not lower semicontinuous, without any luck. Is this actually possible?

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A convex function from ${\mathbb R}^n$ to $\mathbb R$ is always continuous, not just lower semicontinuous. To get examples that are not lower semicontinuous you must allow values of $+\infty$. For example, with $n=1$, $f(x) = +\infty$ for $x \le 0$, $0$ for $x > 0$.