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

Question

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: $$ \|f\|:= \max_{|\beta|<k}\left[ \sup_{x \in \mathbb{R}^n} |(D^{\beta}f)(x)| + \sup_{x,y;\, x\neq y} \frac{|(D^{\beta}f)(x)-(D^{\beta}f)(y)|}{\|x-y\|} \right]<\infty. $$

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 $C^{k,1}$.?

Answer 1

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 $k$.