$2^\omega$ onto the interval $[0,1]$

Question

I want to built a surjective and continuous function of Cantor's set onto $[0,1]$. So, consider the following function: $f:2^{\omega}\rightarrow[0,1]$ such that $f(s):=0,s(0)s(1)s(2)...$

It is clear that this fuction maps $2^{\omega}$ onto the binary representation of elements of $[0,1]$ and it is easy to show that $f$ is continuous. But, I have two problems with this solution

1) As I said above, the image of $f$ is not the interval $[0,1]$. technically $f$ maps onto the binary representations of numbers in $[0,1]$. So, there exists a way the repair my technical issue?

2) There is a formal way to describe the function $f$? i,e, Can $f$ be descibe with a closed formula?

Thanks for advance.

Answer 1

The map is $s\mapsto\sum_{k=0}^\infty 2^{-k-1}s(k)$, just like any $n$-adic expansion would be with $\{0,\cdots,n-1\}^\omega$ and $n$ instead of $2$.