Analysis II Lecture 08 Part 3 sketch of proof of inverse function theorem I

[@yarinmasas2326]

great video! great lecture! formal, intuitive, precise and abstract at the same time. also, very coherent and organized. more videos!!

[@ArthurParzygnat]

Thanks! You got it! Some videos on linear algebra will appear throughout this Spring 2019 semester.

[@speedbird7587]

Hello, You mean that the function f is assumed to be Lipschitz! So the (sufficient condition) inverse function theorem is proven for Lipschitz functions.

[@ArthurParzygnat]

Great observation! The Lipschitz continuity of f is a consequence of the assumptions, so it is not an additional assumption that needs to be made. I should point out that this is slightly different from the the statement of the lemma that I wrote, because the inequality is in the opposite direction! The inequality I wrote is sort of like saying that the inverse is Lipschitz (this is just intuition). As for proving that f is Lipschitz, the open set around the point c can be taken to be enough small (like a ball of some small enough radius) so that its closure (a closed ball) is still in the domain A. The Lipschitz constant can then be obtained by using continuity of the derivative and the fact that the maximum and minimum of any continuous function on a compact domain (the closed ball in this case) is always attained. I'm leaving out some details, which I hope you can fill in---it's a great exercise! You can also check out the details in Theorem 8.11 (page 76) of arthurparzygnat.com/wp-content/uploads/2019/12/3151Spring2017Notes.pdf if you want to see them.