examples with not periodic functions

I'm looking for examples of non-periodic functions $f\in C^{\infty}$ satifying $|f^{(n)}(x)|\leq 1 \forall n\in \mathbb{N},\forall x\in\mathbb{R}$

I know only examples with periodic functions as sin(x) or cos(x)

edit Thank you so much . If I suppose moreover that there exists $a\in\mathbb{R}$ such that $f′(a)=1$, I think that f is necessarily defined by $f(x)=sin(x−a)$

1 Answer

What about $$ \frac{1}{2}\left(\sin\frac{x}{2}+\sin\frac{x}{\pi}\right) $$ ? That is not a periodic function, and every derivative is clearly bounded by $1$ in absolute value.

$$ \sum_{n\geq 2}\frac{1}{n^2}\sin\left(\frac{x}{n^2}\right) $$ works just as well.