# A function that is in $C^k_b(\mathbb R)$ but not in $C^{k+1}_b(\mathbb R)$

Is there an example of a family of functions, index by $$k$$, that is in $$C_b^k(\mathbb R)$$ but not in $$C_b^{k+1}(\mathbb R)$$ for arbitrary $$k$$?

$$C_b^k(\mathbb R)$$ is the space of functions with continuous and bounded derivatives up to $$k$$.

• This should be a duplicate? But I did not find the previous one. Sep 5, 2018 at 10:45
• Let $f(x) = 0$ for $x<0$ and $f(x) = \arctan(x)^{k-1}$ for $x>0$ to get an example with bounded derivatives. Sep 5, 2018 at 10:51
• You may want to know that $C^k(\Bbb R)$ is not the space of functions with bounded derivatives up to $k$, but the space of functions with continuous derivatives up to $k$.
– user562983
Sep 5, 2018 at 10:54
• Doesn't $\arctan$ have bounded derivatives for all $k$ (with the bound increasing with $k$)? Sep 5, 2018 at 12:45

Define $f$ by $$f(x) = \begin{cases} (1-x^2)^{k+1} & |x|<1\\0 & |x|\ge1 \end{cases}$$ Then all derivatives of $f$ up to order $k$ are continuous (they are zero at $|x|=1$). But the $k+1$-st derivative does not exist at $|x|=1$.
Take a continuous function with compact support but not differentiable function $f(x)$ and integrate it $k$ times:
Let $g_1(x) = \int_{-\infty}^{x} f(x) dx$ and define: $g_i(x) = \int_{-\infty}^{x} g_{i-1}(x) dx$
$g_k(x)$ is the required function.
$F (x)=|x|^{\lambda }, ~~k<\lambda<k+1$