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Let $p > 1$, $f$ defined on $L^p(\mathbb{R}^n,\mathbb{R})$, locally Lipschitz and concave, with $f(x) = 0$ when $x \geq 0$ a.e. We define, with $q$ dual to $p$, for any $y\in L^q$ , $g(y) := \sup_{x \in L^p} (f(x) - \int x\cdot y)$, having the property that it is finite on $L^{q,\ge 0}$ (which is the set of a.e positive functions), but it is $+\infty$ on $L^{q,\le 0}$. We know that $g$ is convex and lower semi-continuous on $L^q$.

Is $g$ continuous on $L^{q,\ge 0}$ ? Which simple assumption one could add to make it continuous ?

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  • $\begingroup$ A convex function is locally Lipschitz on the interior of its domain. So if g is finite everywhere, it is continuous. $\endgroup$
    – Dirk
    Dec 9, 2019 at 20:43
  • $\begingroup$ But we don't know the interior of its domain, and $g$ is infinite for negative functions $\endgroup$
    – Alfred
    Dec 9, 2019 at 21:47

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