Find all functions $f:R→R$ which are infinitely many times continuous-differentiable on R, where for every x from $R$, there exists a natural number n, such that $f^{(n)}{(x)}=0$.
I tried using the $Contraction$ and the $Fixed$ $Point$ $Theorem$ with $Lagrange's$ $theorem$, but didn't get any satisfactory results. Any kind of help would be appreciated.