Here is Rudin's change of variables Theorem:
My question is this: what are examples where $\varphi$ not being strictly increasing or not mapping the interval $[a,b]$ to $[A,B]$ breaks the theorem? For example, I am considering $\varphi$ a parabola with minimum point in the unit interval. Using Rudin's special case remark ($\alpha(x)=x$ and $\beta(x)=\varphi$) I keep computing $$\int_0^1 f(x)dx=\int_0^1f(\varphi(y))\varphi'(y)dy$$ with various choices of $f$, and I keep getting these integrals are equal. Does this mean $\varphi$ doesn't need such strict conditions?
If we let $a$ be the minimum point this seems surprising since $\varphi$ is only strictly increasing on $[a,1]$.