# Examples of $C^{k,1}$ functions which are not $C^{k+1}$?

I'm currently reading this paper and the authors define the set $$C^{k,1}(\mathbb{R}^n)$$ as consisting of all functions $$f:\mathbb{R}^n\rightarrow \mathbb{R}$$ having $$k$$ derivatives and for which: $$\|f\|:= \max_{|\beta|

However, I'm having trouble understanding non-trivial examples of such functions (e.x.: if $$f$$ is a polynomial of degree $$k-1$$.)

So I ask: What are examples of functions in $$C^{k,1}(\mathbb{R}^n)$$ but not in $$C^{k+1}(\mathbb{R}^n)$$?*

Specifically, I imagine there is some condition to show that "piecewise polynomial functions of degree at-most $$k$$" belong to $$C^{k,1}$$.?

• Perhaps it would be helpful if you cited the paper you are reading.
– efs
Commented May 27, 2021 at 1:12
– ABIM
Commented May 27, 2021 at 1:14
• When $k$ is even: $\frac{1}{1 + |x|^{k+1}}$. Commented May 27, 2021 at 1:37
• Shouldn't that read $|\beta| \le k$ below the maximum? Commented May 27, 2021 at 8:14

For $$n=1$$ and $$k=1$$, let $$f(x)=x^2 \sin(1/x)$$ for real $$x\ne0$$, with $$f(0)=0$$. Then $$f\in C^{k,1}(\mathbb R^n)\setminus C^{k+1}(\mathbb R^n)$$.
It should be easy to extend this example to the other $$n$$ and $$k$$.