Consider a metric space $X$ and a set-valued map $F : X \to \mathbb{R}^{n}$. We define the minimal selection \begin{equation*} m(F(x)) := \arg\min \big\{ \lvert u \rvert : u \in F(x) \big\}, \end{equation*} where $\lvert \, \cdot \, \rvert$ denotes the 2-norm.

The minimal selection theorem in question can be stated as follows [e.g. Corollary 9.3.3 in Aubin and Frankowska]:

*Let $F$ be a continuous (upper and lower semicontinuous) set-valued map from a metric space $X$ to $\mathbb{R}^{n}$ with nonempty closed convex images. Then the minimal selection is continuous.*

My question is if there exist analogous theorems with $m$ replaced by \begin{equation*} m'(F(x)) := \arg\min \big\{ f(x,u) : u \in F(x) \big\}, \end{equation*} where $f$ is a continuous function such that $f(x, \, \cdot \,)$ is strictly convex on $F(x)$?