Differentiability class: Example of maps that are $C^k$ but not $C^{k+1}$.

Question

Is there a classical example for the fact that the differentiability class satify $$ C^{k+1} \subsetneq C^{k} $$ I'm interested in the $C^{k+1} \neq C^{k}$, then is I'm looking for a classical

Example of maps that are $C^k$ but not $C^{k+1}$. Maps $f_j:\mathbb{R}\to\mathbb{R}$.

For a compact interval the following example should work $$ f_k:[0,2]\to\mathbb{R},\quad f(x)=\begin{cases} 0&\text{for $x\in[0,1)$}\\ 1&\text{for $x\in[1,2]$} \end{cases} $$ then I build recursively $f_{k-1}$ by integrating $f_k$ thus the obtained $f_0$ or $f_1$ is $C^k$ but not $C^{k+1}$.

See my answer to Looking for a function $f$ that is $n$-differentiable, but $f^{(n)}$ is not continuous. — Dave L. Renfro, May 28 '19 at 07:34
@DaveL.Renfro Very interesting and elaborate answer! We (the search engine and I) missed it in the search. — Mohamed Ali, May 28 '19 at 07:50

score 2 · Accepted Answer · answered May 28 '19 at 07:26

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Is the Weierstrass function classical enough? It is continuous but nowhere differentiable; if we integrate it $k$ times we obtain a function which is $C^k$ but not $C^{k + 1}$ for any point in its domain!

answered May 28 '19 at 07:26

Robert Lewis

71,180

score 1 · Answer 2 · answered May 28 '19 at 06:50

1

Your example is even not continuous. Set $F_1(x)=\int_0^x |t|dt$ and $F_{k}(x)=\int_0^x F_{k-1}(t)dt.$

answered May 28 '19 at 06:50

user657324

1,863

Although if you shift the indices it doesn't really matter as long as you start with something integrable. – pancini May 28 '19 at 06:52
No, $\int_0^x tdt$ is $\mathcal C^\infty $. And yes, the domain is $\mathbb R$. – user657324 May 28 '19 at 07:12
@ElliotG: I don't understand your comment. My example is a correct one. – user657324 May 28 '19 at 07:13
It's nothing. Just that you could start with a step function first if you wanted. The second iteration of the integral will be $C^1$, etc. – pancini May 28 '19 at 07:14
Elliot's comment is about mine example, I suppose. – Mohamed Ali May 28 '19 at 07:15
@ElliotG: Indeed, thank you :) – user657324 May 28 '19 at 07:17

score 1 · Answer 3 · answered May 28 '19 at 08:51

I don't know if this is a classical example... Still, let $k \in \mathbb{N}$ and consider

$$ \forall x \in \mathbb{R}, \; f(x) = \begin{cases} x^{k+2} \displaystyle \sin\left( \frac{1}{x} \right) & \text{if } x \neq 0, \\[2mm] 0 & \text{otherwise.} \end{cases} $$

Then, $f \in \mathcal{C}^{k}\left( \mathbb{R} \right)$ but $f \notin \mathcal{C}^{k+1}\left( \mathbb{R} \right)$.

score 0 · Answer 4 · answered May 28 '19 at 08:39

0

What about $f_k(x)=\frac{1}{(k+1)!}|x|\cdot x^k$? We have that $f^{(k)}_k(x)=|x|$.

answered May 28 '19 at 08:39

Michael Hoppe

18,103
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Differentiability class: Example of maps that are $C^k$ but not $C^{k+1}$.

4 Answers4