Denote by $C^n[-\infty,+\infty]$ the class of functions which: have finite limits at $\pm \infty$; and are differentiable $n$ times on the line, with all these derivatives bounded. Denote by $C^3_0$ the subclass of $C^3[-\infty,+\infty]$ which have zero second derivative on $\mathbb{R}$. Endow $C^n[-\infty,+\infty]$ with the supremum norm (so that, in particular, $C^3_0$ inherits this norm).
My question is: is $C^3_0$ dense in $C^3[-\infty,+\infty]$ ?
Many thanks for your help.
"Zero second derivative" = linear function (affine, actually). So you are wondering whether you can approximate uniformly arbitrary smooth functions with affine ones. The answer is negative as, clearly, the uniform limit of affine functions is itself affine.
PS: Actually, with your requirement of finite limits at infinity, the class $C^3_0$ trivializes to $\{0\}$.