Let $f:C\mapsto[-\infty,\infty]$ be an extended real valued function, where $C \subset \mathbb{R}^n$, and let $lev_\alpha f=\{x: f(x) \geq \alpha\}$ be the corresponding $\alpha$-superlevel set.
Question 1 Assume $\alpha_0:=\sup_x f(x)<\infty$ and let $\alpha_i \uparrow \alpha_0$. Under which conditions on $f$ it holds that $lev_{\alpha_i}f$ converges to $\arg \max f $ in the Hausdorff metric? I guess continuity would be the easiest sufficient condition (provided that the argument of the maximum in nonempty).
Question 2 Assume now that $\sup_x f(x)=\infty$ and that such an infinite value is attained at some finite points $x_0$ on the boundary of $C$, more precisely $\lim_{x \to x_0}f(x)=\infty$. Let $C_0$ be the class of such point and let $\overline{lev_{\alpha_i}f}$ be the closure of $lev_{\alpha_i}f$ with respect to the Euclidean topology on $\mathbb{R}^n$. Under which conditions $\overline{lev_{\alpha_i}f}$ converges to $C_0$ with respect to the Hausdorff metric as $\alpha_i \to \infty$?