Explicit formula for a reversible function $f: [0,1] \rightarrow \mathbb{R}$ There is a bijection between $[0,1]$ and $\mathbb{R}$ (because they have a same cardinality). Can we write an explicit formula for such a function? (or at least a reversible function $f$ whose domain is $[0,1]$ and its range cover all real numbers?)

Yes.
First there is an explicit bijection between R and (-1,1).
For a bijection between (-1,1) and (-1,1]
map 1 to 1/2, 1/2 to 1/3 and so ad infinitum.
For a bijection between (-1,1] and [-1,1] do simular with -1.
There is a linear transformation between [-1,1] and [0,1].
Put the the four together for the desired result.
