Derivative of a monotone function discontinuous only at rational numbers

Question

Let $$f\left(x\right)=\sum_{n=1}^\infty \frac{\left[nx\right]}{2^n}.$$
where $\left[•\right]$ is the floor function.
I proved that it is convergent for every real number $x$, and $f$ is strictly increasing, continuous at irrational points and discontinuous at rational points.
Since monotonity, $f$ must be differentiable almost everywhere.
I guess it is differentiable at irrational points, and the derivative is $0$.
My question is: Does $f’\left(s\right)$ exists for every irrational number $s$? If it exists, is it equal to $0$?