Does there exist a function$f: [0,1]\rightarrow \mathbb{R}$ almost everywhere $0$ but whose range is equal to $\mathbb{R}$

Question

Does there exist a function$f: [0,1]\rightarrow \mathbb{R}, f=0,a.e$ but whose range is equal to $\mathbb{R}$?

I can't image what this kind function looks like.

Answer 1

Sure, the Cantor set has cardinality $c$ and measure $0$. So there is a bijection between the Cantor set and $\mathbb{R}$. There is even an almost nice description of the bijection. It can be done by a modification of the Cantor function.