Proof that a quocient set is homeomorphic to \mathbb{R}

Question

I'm having trouble solving this:

Consider $X:=\mathbb{R}$ with equivalence relation: $$xRy \iff x=y \text{ or } x,y\in [0,1].$$ Proof that $X/R$ is homeomorphic to $\mathbb{R}$.

How do I write the set $X/R$?

I can't find the funcition for the homeomorphism.

Answer 1

Hint

Let $$ f(x) = \begin{cases} x & x < 0 \\ 0 & 0 \le x \le 1\\ x-1 & x > 1 \end{cases} $$ Then $f$ is continuous, and for every $u \in \Bbb R$, $f^{-1}(\{u\})$ is the equivalence class containing $u$. That should make it easy to find a homeomorphism.