I need to prove the following version of integration by subtitution:
Let $f:[a,b]\to \Bbb R \,$ be an integrable function, and $\phi:[\alpha,\beta]\to[a,b]$ a linear function such that $\phi(\alpha)=a\; ,\phi(\beta)=b$. Then: $$\int^b_af(x)\,dx =\int^\beta_\alpha f(\phi(t))\cdot\phi'(t)\,dt$$
Since I don't know that $f$ if continious, I can't even assume it has a primitive function, and I'm pretty much out of ideas. Any help will be highly appreciated. Thanks!
Eg [x] is not a continuous function still we can find its definite integral .Function is given to be linear so its continuous.. For your function let $\phi (t)=x $ thus $\phi '(t)dt=dx $ now also its given that $\phi ^{-1}(a)=\alpha,\phi ^{-1}(b)=\beta $ thus the function $\int _a ^b f (x)dx=\int _{\alpha} ^{\beta} f (\phi (t)).\phi '(t)dt $