As already discussed the answer to the question, as stated, is NO. However, one can get close if one is willing to go to a measure theoretic statement. For example, the statement $$ |\{x \in [a,b]:\quad |f'(x)| < \varepsilon\}| \to 0 $$ as $\varepsilon \to 0$, is an easy consequence of continuity. However, one can quantify this. If I remember correctly, it is relatively easy to prove $$ |\{x \in [a,b]:\quad |f'(x)| < \varepsilon\}| \leq C \varepsilon^{1/d} $$ where $C$ is an (universal) constant and $d$ the degree of the polynomial $f$. I think one can give a proof by just rescaling Polya's inequality.