IR-TST 2022 Problem 12

suppose that $A$ is the set of all Closed intervals $[a,b] \subset \mathbb{R}$ . Find all functions $f:\mathbb{R} \rightarrow A$ such that $\bullet$ $x \in f(y) \Leftrightarrow y \in f(x)$ $\bullet$ $|x-y|>2 \Leftrightarrow f(x) \cap f(y)=\varnothing$ $\bullet$ For all real numbers $0\leq r\leq 1$ , $f(r)=[r^2-1,r^2+1]$ Proposed by Matin Yousefi