Compact set / Multifunction

Question

Let $K(.)$ a multifunction defined on $\mathbb{R}$ defined as follows: $$K(x):=\begin{cases} K=\{0\} \text{ if } x\neq0,\\ K=[-1,1] \text{ if } x=0. \end{cases}$$

Can we deduce directly that for all $x\in \mathbb{R}$ $K(x)$ is a compact of $\mathbb{R}$ since $\{0\}$ and $[-1,1]$ are both compacts of $\mathbb{R}$?

Answer 1

$\{0\}$ and $[-1,1]$ are compact subsets of $ \mathbb R.$ Hence $K(x)$ is compact for each $x$.