Let $(X ,d)$ be a complete metric space. $A :X \rightarrow 2^{\mathbb{R}}$ is a multivalued function (i.e., $A(x) \subseteq \mathbb{R}$ and $A(x)$ is non-empty for all $x \in X$).
Suppose that $A(x)$ is compact for all $x \in X$ and that $A$ is upper hemicontinuous.
Proof or find a counter example: $A$ has a continuous selection.
Proof: Define $k(x) =\max A(x)$. Since $A(x)$ is compact, non-empty, and $A(x) \subseteq \mathbb{R}$, $k(x)$ is well defined. We need to show that $k(x)$ is continuous.
Let $x_{n} \rightarrow x$ and consider the sequence $\{k(x_{n})\}$ where $k(x_{n}) \in A(x_{n})$ for each $n$. I am not sure how to continue.