HMMT 2015 T7

Let $f:[0,1] \rightarrow \mathbb{C}$ be a nonconstant complex-valued function on the real interval $[0,1]$. Prove that there exists $\epsilon>0$ (possibly depending on $f$ ) such that for any polynomial $P$ with complex coefficients, there exists a complex number $z$ with $|z| \leq 1$ such that $|f(|z|)-P(z)| \geq \epsilon$.