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 Catalan mathematicians:

F. Sunyer i 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. Sunyer i 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?