Prove that $ f(x)=\sum\limits_{n=3}^{\infty} \frac{\min\limits_{k\in Z} |4^n \cdot x - k|}{4^n} $ is continuous and is not differentiable.

I have this exercise from elementary analysis:

Define the function $f: \mathbb R \to\mathbb R$ by$$ f(x)=\sum\limits_{n=3}^{\infty} \frac{\min\limits_{k\in Z} |4^n \cdot x - k|}{4^n} $$Prove that(1) $f$ is continuousand(2) $f$ is not differentiable at any $x \in\mathbb R$

I noticed that this may have some similarity with the Blancmange curve , and I think I could use the M-Test to prove the continuity but I have no clue how to prove that is not differentiable at any $x$.