Pringsheim's Theorem

Let $C^\omega(I)$ be the set of real Analytic Functions on $I$. Then $C^\omega(I)$ is a Subalgebra of $C^\infty(I)$. A Necessary and Sufficient condition for a function $f\in C^\infty(I)$ to belong to $C^\omega(I)$ is that

$$\vert f^{(n)}(x)\vert\leq k^n n!$$

for $n=0$, 1, ... for a suitable constant $k$.

See also Analytic Function, Subalgebra

References

Iyanaga, S. and Kawada, Y. (Eds.). Encyclopedic Dictionary of Mathematics. Cambridge, MA: MIT Press, p. 207, 1980.