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 $\epsilon$?


No. Consider the very simple upper hemi-continuous correspondence $\phi$ from $[0,1]$ to $[0,1]$ such that
$$ \phi(x)= \begin{cases} \{0\} \text{ if }x<1/2\\ \{0,1\} \text{ if }x=1/2\\ \{1\} \text{ if }x>1/2. \end{cases} $$ Clearly, $[0,1]$ is an ANR. The correspondence $\phi$ looks almost like a discontinuous function, and that is why it cannot be well approximated by a continuous function. If there were an $\epsilon$-approximation $f_\epsilon$ with $\epsilon<1/4$, there must be some $x$ with $f_\epsilon(x)=1/2$ and there we have $d\big(f_\epsilon(x),\phi(x)\big)=1/2>\epsilon$.

It should be pointed out that almost the same argument works if we were to take $\phi(1/2)=[0,1]$. In that case, $\phi$ would be convex-valued but still no $\epsilon$-approximation as defined above possible. That is why one usually takes the difference between the graphs of the correspondence and the function as the relevant notion of approximation. The counterexample above is robust to this change.

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