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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

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

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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.

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