The number of components of preimage of a continuum under a polynomial

Question

Given a ploynomial $f$ with degree $d$, then we have a dynamical system $(\mathbb{C},f)$, the critical points are those points $z$ such that the derivative of $f$ at which is zero. If we have a proper continuum $N\subset\mathbb{C}$, then we know that the set $f^{-1}(N)$ have at most $d$ components with each component maps onto $N$. If further, we assume the continuum $N$ is full(which means that $N$ and its complement are both connected) and does not contains the image of critical points(we call critical valve), can we say that there are exactly $d$ components of $f^{-1}(N)$, and $f$ is injective when f is restricted on each component?