The Inverse Function Rule
Ifis continuous and one to one on an intervalthenexists and is continuous. Ifis differentiable then the following theorem is applicable:
Supposeis continuous and differentiable withfor allThenis one to one,is continuous and differentiable onandfor all
Proof: Sincefor allthenis one to one.
Suppose that–is continuous sois a closed interval. Chooseand any sequence inconverging toLet foris continuous sois continuous andconverges to and sinceis one to onefor allSinceis differentiableconverges toand sinceis one to one forHenceconverges toThusis differentiable and