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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!

nikibiki
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  • If g' is continuous and is never 0 then if g' has a negative and a positive value somewhere g' would also take all the values between these 2 values, including 0. – Lelouch Jan 12 '22 at 15:24
  • Darboux's theorem promises that any derivative satisfies the intermediate value theorem. (So, while not all derivatives are continuous, all are close enough for the IVT to work.) So, if a derivative is positive somewhere on $I_0$ and negative somewhere else, it is zero at least once between them. Consequence: if $g'$ is never zero on $I_0$, $g'$ can never change sign on $I_0$. – Eric Towers Jan 12 '22 at 15:27
  • @EricTowers thank u for the answer. May be I formulated the question bad or I didn't understand, but how we now, that $g'\neq0$ is? We have to prove it. And the answer is that IVT says it, but I don't understand it. – nikibiki Jan 12 '22 at 15:36

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