Denote by Cn[−∞,+∞] the class of functions which: have finite limits at ±∞; and are differentiable n times on the line, with all these derivatives bounded. Denote by C30 the subclass of C3[−∞,+∞] which have zero second derivative on R. Endow Cn[−∞,+∞] with the supremum norm (so that, in particular, C30 inherits this norm).
My question is: is C30 dense in C3[−∞,+∞] ?
Many thanks for your help.
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