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 ?