Example of a diffeomorphism of class $C^{k}$ which is not $C^{k+1}$

Question

Can anyone give me an example of a map $f:\mathbb{R}\to\mathbb{R}$, which is a diffeomorphism of class $C^{k}$ but it is not a diffeomorphism of class $C^{k+1}$?

Answer 1

The Cantor staircase function is an example of a function which is continuous but not differentiable ($C^0$ but not $C^1$) from the unit interval to the unit interval.

Let's call that $u_0(x)$. In general, we will construct $u_k(x)$ to be a function which is $C^k$ but not $C^{k+1}$ from the unit interval onto itself, which is monotone. We do this by setting $$ u_{k+1}(x) = \frac{\int_0^x u_k(s) ds}{\int_0^1 u_k(s) ds}$$ In general, $u_k$ is 1-1 and onto for $k>1$, since it is absolutely continuous with a derivative that is positive except at one point.

Now the tangent function is a nice (1-1, onto, monotone, smooth) map from the interval $[-\frac{\pi}{2},\frac{\pi}{2}]$ to $\mathbb{R}$. So setting $$f_k(x) = \tan \left(u_k\left(\frac{\arctan(x)}{\pi} + \frac{1}{2}\right)\right)$$ should give you the function you're looking for.

Answer 2

Start out with a sample path from a continuous Brownian motion that is positive. It is continuous but nowhere differentiable. Since it is continuous one may integrate it to obtain a differentiable map whose derivative is not differentiable (by the Fundamental Theorem of Calculus). Since it is positive it integral is an increasing function. Do this k times.