Let $f(x)$ be a function and $$\mathfrak{F}:C^{\infty}\rightarrow C^{\infty} : f(x)\mapsto\lim_{n\rightarrow\infty}\frac{d^nf}{dx^n}$$ when it exists. Then what is the domain of $\mathfrak{F}$?
I know that functions $g(x)=ae^x$ are in the domain and that polynomials are there too but does it contain anything else?
Edit : I suppose the norm in $C^\infty$ can be taken as $$\parallel f\parallel_{C^\infty}=\sup_{x\in\mathbb{R}}\;\sup_{k\in\mathbb{N}}\;\left|\frac{d^kf}{dx^k}\right|$$