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 $C^{k,1}$.?

  • $\begingroup$ Perhaps it would be helpful if you cited the paper you are reading. $\endgroup$
    – efs
    Commented May 27, 2021 at 1:12
  • $\begingroup$ @EFinat-S I added the paper link. $\endgroup$
    – ABIM
    Commented May 27, 2021 at 1:14
  • 1
    $\begingroup$ When $k$ is even: $\frac{1}{1 + |x|^{k+1}}$. $\endgroup$ Commented May 27, 2021 at 1:37
  • $\begingroup$ Shouldn't that read $|\beta| \le k$ below the maximum? $\endgroup$
    – user56029
    Commented May 27, 2021 at 8:14

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 $k$.

  • $\begingroup$ Thanks, Im wondering if this set contains some sort of "higher-order" splines... $\endgroup$
    – ABIM
    Commented May 27, 2021 at 1:16

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