Prove the continuity

Question

$f(x): [0, 1) \rightarrow \mathbb{R}$. Prove that the function $f(x) = \sum_{n=1}^{\infty}2^{-n}\{\frac{[2^nx]}{2}\}$ is continuous. Please, give me a hint where to start (I want to prove it using definition with $\varepsilon$ and $\delta$) P.S. {x} is fractional part of x and [x] is integer part of x. I have $|x - y| \le \delta$ and I need to tranform this thing $|\sum_{n = 1}^{\infty}(2^{-n}(\{\frac{[2^nx]}{2}\} - \{\frac{[2^ny]}{2}\}))|$ somehow to show that this is less than $\varepsilon$ for some $\delta$. I dont know how to get rid of $\{\frac{[2^nx]}{2}\} - \{\frac{[2^ny]}{2}\}$