# Show that the inverse function $f^{-1} : f(I) \rightarrow I$ is $k$-times continuously differentiable.

Let $k \in \mathbb{N}$. Let $I \subset \mathbb{R}$ be an open interval. Let $f : I \rightarrow \mathbb{R}$ be a $k$ times continuously differentiable function with $f'(x) \not= 0$ for all $x \in I$.

Show that:

$1)$ $f$ is injective.

2) $f(I)$ is an open interval.

3) the inverse function $f^{-1} : f(I) \rightarrow I$ is $k$-times continuously differentiable.

So $1)$ and $2$) weren't a problem. but I need help with $3)$.

I've already shown: $1)$, $2)$, $f^{-1}$ is continuous and $f^{-1}$ is differentiable.

Many thanks in advance

• @Arthur, your $f$ is only $C^1$. $f$ is $C^k$ (see the title). – Martín-Blas Pérez Pinilla Apr 12 '18 at 12:40
• @Martín-BlasPérezPinilla I missed the title. Gotcha. – Arthur Apr 12 '18 at 12:41
• @Arthur, body edited. – Martín-Blas Pérez Pinilla Apr 12 '18 at 12:41

Let me use $g=f^{-1}$, for ease of notation. By definition, for every $x\in f(I)$, $$f(g(x))=x$$ Differentiating both sides gives $$f'(g(x))g'(x)=1$$ and therefore $$g'(x)=\frac{1}{f'(g(x))}$$ proving that $g'$ is continuous. Suppose $k\ge2$: then we can go further with $$g''(x)=-\frac{f''(g(x))g'(x)}{(f'(g(x))^2}$$ proving that $g''$ is continuous.
Going on this way would be complicated. However, we can use Leibniz's formula: if $F$ and $G$ are $n$ times continuously differentiable functions, then $$D^n(FG)=\sum_{i=0}^n\binom{n}{i}D^iF\,D^{n-i}G$$ and we can use $F(x)=f'(g(x))$, $G(x)=g'(x)$, so for $1\le n\le k-1$, $$0=D^n(FG)=\sum_{i=0}^n\binom{n}{i}D^iF\,D^{n-i}G= g^{(n+1)}+\sum_{i=1}^n\binom{n}{i}D^iF\,D^{n-i}G$$ and it's a matter of showing that $D^iF$ only depends on derivatives of $f$ up to $i+1$ and of derivatives of $g$ up to $i$, so $$g^{(n+1)}=-\sum_{i=1}^n\binom{n}{i}D^iF\,D^{n-i}G$$ only depends on derivatives of $f$ up to order $n$ and of derivatives of $g$ up to order $n$.
Case $k = 1$: $f^{-1}$ is monotonic $\implies$ only can have jump discontinuities. But if the image of an open interval an open interval, no jump discontinuity is possible.