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

Question

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 $k$.

Answer 1

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 $|x|=1$.

Answer 2

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: $g_i(x) = \int_{-\infty}^{x} g_{i-1}(x) dx$

Use : Lebesgue's Differentiation Theorem for Continuous Functions to prove that the derivatives exist.

$g_k(x)$ is the required function.

Answer 3

$F (x)=|x|^{\lambda }, ~~k<\lambda<k+1$