Originally Posted by

**Tahoe** I donīt want you to solve me the thing but when you look at a problem for way too long you get more and more stuck and that is why I kindly ask you for some help.

I want to sum up my current thoughts:

$\displaystyle \lim\limits_{x \rightarrow 2^{-i}k} \frac{f_{i} \left(x \right) - f_{i} \left(2^{-i}k \right)}{x - 2^{-i}k} = \lim\limits_{x \rightarrow 2^{-i}k} \frac{\int_{0}^{x} \left(-1 \right)^{\lfloor 2^{i} \cdot t \rfloor} \ dt - \int_{0}^{2^{-i}k} \left(-1 \right)^{\lfloor 2^{i} \cdot t \rfloor} \ dt}{x - 2^{-i}k} \\ = \lim\limits_{x \rightarrow 2^{-i}k} \frac{ \int_{x}^{2^{-i}k} \left(-1 \right)^{\lfloor 2^{i} \cdot t \rfloor} \ dt }{x - 2^{-i}k}. $