Let us consider the cubic equation
$$a(t)x³+b(t) x²+c (t)x+d(t)=0\qquad (*)$$
where $a(t),b(t),c (t),d(t)$ are rational functions defined in the whole real line minus some finite set of points. We know that (*) has still a real root $x(t)$ given in function of the coefficients (or in function of $t$).
My question is: How one can prove that the range (the codomain) of the real solution $x(t)$ is also the whole real line minus a finite set of points?
Maybe this link may help: Continuity of the roots of a polynomial in terms of its coefficients