The theorem:

Theorem: Let $f(x)$ be $C^\infty$ on $(c,d)$ such that for every point $x$ in the 
interval there exists an integer $N_x$ for which $f^{(N_x)}(x)=0$; then $f(x)$ 
is a polynomial.

is due to two Spanish mathematicians:

F. S. Balaguer E. Corominas, Sur des conditions pour qu'une fonction infiniment dérivable soit un polynôme. Comptes Rendues Acad. Sci. Paris, 238 (1954), 558-559.

F. S. Balaguer E. Corominas, Condiciones para que una función infinitamente derivable sea un polinomio. Rev. Mat. Hispano Americana, (4), 14 (1954).

The proof can also be found in the book (p. 53):

W. F. Donoghue, Distributions and Fourier Transforms, Academic Press, New York, 1969.

I will never forget it because  in an "Exercise" of the "Opposition" to 
became "Full Professor" I was posed the following problem:

What are the real  functions  indefinitely differentiable on an interval such that 
a derivative vanish at each point?