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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?

3lectrologos
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1 Answers1

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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 $x > 0$.

Robert Israel
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  • So saying that a function is convex and lower semicontinuous is equivalent of saying that the function is convex and continuous. Is that right? – user693 Jun 08 '13 at 13:56
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    Careful. If there are no infinite values then convex & lsc implies continuous. Otherwise the above is a counterexample. – Christian Bueno Apr 22 '20 at 19:16
  • Can you tell something about the infinite-dimensional case? – rod Oct 30 '23 at 17:42
  • A lower semicontinuous real-valued convex function on an open convex subset of a Banach space is continuous. – Robert Israel Oct 30 '23 at 22:02