Prove inverse function k times differentiable

Asked Apr 20 '21 at 08:21

Active Apr 20 '21 at 08:21

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Given two functions $f,g$ $k$ times differentiable, prove that their composition is $k$ times differentiable.

Let $f$ be an injective function, show that its inverse is $k$ times differentiable.

Looking at their first derivative expression, I can get a sense of how these are true. I guess induction would be the way, but I am not sure how to write the proof. Do I need to write out the n-derivative formula for these and then prove the n+1 case is correct, or a simpler argument is sufficient?

asked Apr 20 '21 at 08:21

sucksatmath

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See https://math.stackexchange.com/q/3870623/42969 for the inverse function part. – Martin R Apr 20 '21 at 08:31
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Note that you have to require $f'(x) \ne 0$, otherwise $f(x) = x^3$ is a counteexample. – Martin R Apr 20 '21 at 08:32

Prove inverse function k times differentiable

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