I'd like to know if there are any known-results on the existence of continuous approximation theorems for upper hemicontinuous (aka upper semicontinuous) maps $\phi: X\rightarrow Y$ which are finite valued. There are a number of such results, perhaps most famously the Granas–Górniewicz–Kryszewski (G-G-K) theorem, when $X$ is an ANR (absolute neighborhood retract) and each $\phi(x)\subset Y$ is compact and satisfies a (generalized) contractibility condition. Such approximations are important in establishing fixed point theorems, such as that of Katukani, for upper hemicontinuous maps from $X$ to itself (i.e., the existence of some $x\in X$ such that $x\in \phi(x)$ in the case $X=Y$) which generalize the classical fixed point theorems, such as that of Brouwer's, in the continuous (single-valued) case.
Before asking my question, I'll state the relevant definitions and theorems.
$\textbf{Definition 1.}$ A multivalued-function $\phi: X\rightarrow Y$ is a correspondence such that $\phi(x)\subset Y$ for all $x\in X$, $\phi(x)\neq\emptyset$. It is upper hemicontinuous if for any $x\in X$ and any open neighborhood $V$ of $Y$ containing $\phi(x)$, there is an open neighborhood $U$ of $x$ such that $\phi(U)\subseteq V$.
$\textbf{Definition 2.}$ For $\epsilon>0$, an $\epsilon$-approximation $f_\epsilon: X\rightarrow Y$ of a multi-valued map $\phi: X\rightarrow Y$ is a continuous (single-valued) map such that $d(f_\epsilon(x), \phi(x))<\epsilon$ for all $x\in X$.
There are some theorems that guarantee that $\phi$ can be $\epsilon$-approximated for any $\epsilon>0$. For instance, suppose that
(1) $X$ is a compact absolute neighborhood retract ANR (or $X$ is an ANR and each $\phi(x)$ is compact-valued) and that
(2) Each $\phi(x)\subset Y$ is convex, or more generally contractible, or satisfies a generalized contractibility such as (a) $\phi(x)$ is contractible in $Y$, or, as, in the G-G-K theorem, even if (b) each open neighborhood $U$ of $\phi(x)$ contains an open neighborhood $V\subseteq U$ of $\phi(x)$ so that $V$ is contractible in $U$ (this is called "proximal contractibility").
Then $\phi: X\rightarrow Y$ has an $\epsilon$-approximation for any $\epsilon>0$.
Finally, my question:
$\textbf{Question}.$ Suppose we have that $\phi: X\rightarrow Y$ satisfies (1), but if instead of (2) we know that each $\phi(x)$ is a finite disjoint union of such sets, e.g., if each $\phi(x)$ is finite (but $|\phi(x)|$ needn't be constant). Can we still guarantee an $\epsilon$-approximation for any $\epsilon$?
