Hi!

I got the following:

A function $\displaystyle f_{i} \left(x \right) = \int_{0}^{x} (-1)^{\lfloor t \cdot 2^{i}\rfloor} \ dt$ and the binary expansions with $\displaystyle \tilde{b_{i}} = 1 - b_{i}$ which are given by

$\displaystyle x_{0}= 0.....0b_{n+1} b_{n+2} \dots$ and

$\displaystyle x_{1}= 0.....0\tilde{b_{n+1}} \tilde{b_{n+2}} \dots$

I guess that the function $\displaystyle f_{i} \left(x \right)$ is reflection symmetric about the vertical line $\displaystyle x=2^{-(n+1)}, n \geq 1$.

Is the reflection imagine about that line of $\displaystyle x_{0}$ $\displaystyle x_{1}$ because I can verify that

if $\displaystyle x_{0} < x_{1}$ then I get $\displaystyle x_{0} + 2 (x - x_{0}) = x_{1}$ and

if $\displaystyle x_{1} < x_{0}$ then I get $\displaystyle x_{1} + 2 (x - x_{1}) = x_{0}$ ?

Thanks.