Proof of integral substitution rule

Question

I have the proof of the integral substitution rule at the university. In order this rule to use, I must have some conditions. So $ f: I\to R$ and $g: I_0 \to I$ and $I,I_0$ are not trivial intervals and $f,g$ are continuous and diffbars functions with 1.)$\forall t \in I_0 \quad g'(t)\neq0$. Our Professor says, that we can use Intermediate value theorem and so we know, that g'(t) is positiv or negativ everywhere. My question is WHY? Why do we know that? How I.value theorem implies it? Can somebody explain it? Thank u!