can we prove that a certain supremum of affine functions is frechet differentiable or at least continuous?

Question

Let $X$ be a Hilbert space. Let $A\colon \operatorname{dom} A\to X$ be linear operator with closed graph (not necessarily bounded). Define $$ g\colon \operatorname{dom}A\to \mathbb{R}:x\mapsto \sup_{y\in \operatorname{dom}A}\{<Ax,y>-<Ay,y>\}. $$ If $\forall x\in \operatorname{dom}A$ we have $g(x)<+\infty$. Can we prove that $g$ is Fréchet-differentiable (or at least continuous) on $\operatorname{dom}A$.