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The epigraph of a function $f:\mathbb{R}^{n}\to [-\infty,+\infty]$ is the set of points $(x,\mu)\in\mathbb{R}^{n+1}$ satisfying $f(x)\leq \mu$, and is denoted by $\mathrm{epi}(f)$. I somehow got convinced that any such function can be recovered from its epigraph via the formula $f(x)=\inf\{\mu|(x,\mu)\in \mathrm{epi}(f)\}$ (mainly because of its geometric intuition), but now I have second thoughts. Is this formula true?

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Yes. Consider $\mathrm{epi}(f)\cap(\{x\}\times\mathbb R)$. –  Did Nov 13 '11 at 23:24
    
Perhaps it is useful to remind that $\inf\emptyset=+\infty$. As an exercise in using google I tried to find it on google books and I found this: Fundamentals of convex analysis By Jean-Baptiste Hiriart-Urruty, Claude Lemaréchal, p.76 –  Martin Sleziak Nov 16 '11 at 11:47

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