Let $(X,d_X)$ and $(Y,d_Y)$ be metric spaces, give $X\times Y$ the square metric $d=max\{d_X,d_Y\}$, and let $\mathscr{H}$ denote the nonempty closed, bounded subsets of $X\times Y$. The metric $d$ induces the Hausdorff metric $\overline{d}$ on $\mathscr{H}$. Consider $C(X,Y)$ under the uniform metric. Let $gr:C(X,Y) \rightarrow \mathscr{H}$ send $f\in C(X,Y)$ to its graph $G_f =\{ (x, f(x))\ |\ x\in X\}$.
(a) Show that $gr$ is injective and uniformly continuous.
(b) Let $\mathscr{H}_0 =gr(C(X,Y))$, $g : C(X,Y) \rightarrow \mathscr{H}_0$. If $f\in C(X,Y)$ is uniformly continuous, then $g^{-1}$ is continuous at $f$.
(c) Give an example where $g^{-1}$ is not continuous at $f$.
(d) If $X$ is compact, show that $gr$ is an embedding.
I'm not sure how to approach this problem. For one, if we let $X,Y=\mathbb{R}$, then $G_{x^2}$ doesn't seem to be bounded with respect to $d$.