Piecewise continuous selection from a correspondence Take a correspondence $C:[0,1] \rightarrow [0,1]$ which is non-empty, convex valued and has closed graph.
For each $x \in [0,1]$ let $a(x) = \{ \max \ y\in C(x), \min \ y \in C(x) \}$ (i.e., the set $a(x)$ contains two points only: the minimum value taken by the correspondence and the maximum). (also remember that $C$ is convex valued).
Now consider $\bigcup_x a(x) $ and assume that this takes up only a finite number of real values.
Is then possible to find a finite set $\{x_1,\dots,x_n\} \subset [0,1]^k$ such that for all $y\in [0,1]$ there exists $x_i$ such that $y \in C(x_i)$?
Note that it is important that the finite set  $\{x_1,\dots,x_n\}$ span the entire range of the correspondence.
Thanks
 A: Choose any $x$ in $[0,1]$ and let
$$\min C(x)=\alpha_1,\quad\quad \max C(x)=\alpha_2.$$
Note that the minimum and maximum exist because $C(x)$ is a closed and bounded subset of $\mathbb{R}$. Since $C(x)$ is convex, we have that for any $0\leq\theta\leq1$
$$\theta \alpha_1+(1-\theta)\alpha_2\in C(x).$$
The above implies that $C(x)$ is the interval $[\alpha_1,\alpha_2]$. In other words, for any $x$ you pick, $C(x)$ is a closed interval. Now, suppose that
$$\bigcup_{x\in[0,1]}a(x):=\{\alpha_1,\alpha_2,\dots,\alpha_n\}$$
where $\alpha_1<\alpha_2<\dots<\alpha_n$. There are only $\begin{pmatrix}n \\ 2\end{pmatrix}$ intervals we can form by choosing end points out of $\{\alpha_1,\alpha_2,\dots,\alpha_n\}$, namely
$$[\alpha_1,\alpha_2],[\alpha_1,\alpha_3],\dots,[\alpha_1,\alpha_n],[\alpha_2,\alpha_3],\dots,[\alpha_{n-1},\alpha_{n}].$$
Hence, by picking the $x_i$s such that 
$$C(x_1)=[\alpha_1,\alpha_2],\quad C(x_2)=[\alpha_1,\alpha_3],\quad\dots,\quad C\left(x_{(n\text{ choose }2)}\right)=[\alpha_{n-1},\alpha_n],$$ 
you have your desired set.
ASIDE: By the way, $C:[0,1]\to 2^{[0,1]}$, that is, $C$ maps from $[0,1]$ to the power set of $[0,1]$. If $C:[0,1]\to[0,1]$, then $C$ is a function and not a correspondence (or multivalued function or set-valued function).
