Can we prove that polynomial functions can always be split into bijective restrictions?

The derivative of a polynomial function is a polynomial function, and as such, it is a continuous function with only finitely many zeros. Those zeros divide the line into finitely many intervals. On the interior of each of those intervals, the derivative is either everywhere positive or everywhere negative. (That follows from the intermediate-value theorem, which says a continuous function never changes sign in an interval in which it is nowhere zero.)

If the derivative of a continuous function is everywhere positive or everywhere negative on an open interval, then the function is strictly monotone, and therefore bijective, on the corresponding closed interval.