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

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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 $\{0\}.$

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  • $\begingroup$ Thank you szw1710. Can we add more assumptions to have the result? $\endgroup$ – user2015 Apr 17 '17 at 19:13
  • $\begingroup$ 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. $\endgroup$ – szw1710 Apr 17 '17 at 19:22
  • $\begingroup$ Thank you szw1710. It was a comment from my professor. I think it was just an assumption he made. $\endgroup$ – user2015 Apr 17 '17 at 19:35
  • $\begingroup$ 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 $x_0$. $\endgroup$ – szw1710 Apr 17 '17 at 19:39
  • $\begingroup$ Nice observation szw1710. Thanks. $\endgroup$ – user2015 Apr 17 '17 at 20:13

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