Hemicontinuous compact-valued mapping on $\mathbb{R}$ with no continous selection

Question

I was told that there exists a mapping $f \colon \mathbb{R} \to \mathcal{P}(\mathbb{R})$ such that

• $f(x)$ is non-empty and compact for every $x \in \mathbb{R}$,
• $f$ is both upper and lower hemicontinuous,
• there is no continuous selection of $f$.

Such a function would serve as an example showing that Michael's selection theorem fails when we omit the convexity assumption, even if we assume in return that the set-valued mapping is upper hemicontinuous and has compact values.

Does anybody know hot to construct the mapping $f$?

I'll appreciate any help.

Answer 1

I beleive I got it. The mapping $f \colon \mathbb{R} \to \mathcal{P} (\mathbb{R})$ given by $f(x)=[-1,1]$ for $x \leq 0$ and by $f(x)= [-1,1] \setminus \Big\lbrace t \in \mathbb{R} \, ; \ \big|t-\sin \frac{1}{x} \big|< \frac{x}{x+1} \Big\rbrace$ for $x > 0$ seems to work fine.