A continuous function with bounded variation which isn't monotonic.

Question

so I got a homework assignment to prove a function is continuous with bounded variation but isn't monotonic on any subinterval. The function is defined as, let: $$g\left(x\right)=\left|x\right|\ x\in\left[-\frac{1}{2},\frac{1}{2}\right]$$and than $g\left(x\right)=g\left(x+1\right)$ and $g_{n}\left(x\right)$ to be the minimum of $\left\{ g\left(x\right),8^{-n}\right\}$. I proved that the series: $$\sum_{n=1}^{\infty}2^{-n}g_{n}\left(8^{n}x\right)$$ converging for each real number. and so the function: $$f\left(x\right)=\sum_{n=1}^{\infty}2^{-n}g_{n}\left(8^{n}x\right)$$ is continuous. But I can't manage to prove that the function has bounded variation, Any hints\help would be great.