real analysis - Approximation of continuous functions by a functions with vanishing second derivative

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.