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?