Proof that $f : \mathbb{R} \rightarrow [0,1]$ with $f(x) = x−\lfloor x\rfloor$ is bijective

Question

Prove that $f : \mathbb{R} \rightarrow [0,1]$ with $f(x) = x−\lfloor x\rfloor$ is bijective

$\lfloor x\rfloor$ is the floor function of $x$. I don't know how to work with floor functions first of all. I also don't know how to start proving surjectivity and injectivity

Answer 1

$$f(1)=0$$

and

$$f(2)=0$$

Can you conclude whether it is a bijection?

Answer 2

It's hard to prove a false statement. ;-)

The function is not injective, because $f(n)=0$ for every integer $n$.

The function is not surjective, because $f(x)<1$, for every real $x$.