Problem in Putnum competition?

Question

Suppose $f:\mathbb{R}\longrightarrow \mathbb{R}$ is a continuous function and $f(2x^2 -1)=2xf(x)$ for all $x\in \mathbb{R}$.

Prove $f(x)=0\,\,\text{for all} \, x\in [-1,1].$