I'm looking for a condition of the existence of continuous approximate selection for a (multi-valued) minimiser mapping.
That is, let's say a continuous function $f: X \times U \to \mathbb{R}$ is given for $X \subset \mathbb{R}^{n}$ nonempty compact and $U \subset \mathbb{R}^{m}$ nonempty convex compact.
The (multi-valued) minimiser mapping $G: X \to \mathbb{R}$ is defined as $G(x) = \arg\min_{u \in U} f(x, u)$. I think $G$ is upper hemi-continuous by Berge's theorem.
Now, for given $\epsilon > 0$, I'd like to provide a characteristic of $f$ (or, $G$) so that there exists a continuous $\epsilon$-approximate selection $g_{\epsilon}$ of $G$, i.e., $\text{Graph}(g_{\epsilon}) \subset B(\text{Graph}(G), \epsilon)$.
I found several theorems about this topic, but mostly they require that $G$ be convex-valued, for example, [1, Theorem 9.2.1]. However, I can easily find an example without the convexity. For example, for $f$, defined on a closed interval $U$ and whose value does not vary along the axis of $X$, has distinct two minima $u_{1}^{*} \neq u_{2}^{*}$ on $U$, $G$ is not convex valued (in fact, $G(x) = \{u_{1}^{*}, u_{2}^{*}\}$ for all $x \in X$) but there exists a continuous (approximate) selection $g(x) = u_{1}^{*}$.
Cited works
[1] J.-P. Aubin and H. Frankowska, Set-Valued Analysis. Boston, MA: Birkhäuser Boston, 2009. doi: 10.1007/978-0-8176-4848-0.