= Factorial derivative criterion for real analyticity
{title2=$|f^{(n)}|\leq CA^n n!$}
A <smooth function> on a real open interval is <real analytic> exactly when, near each point, there are constants $C,A>0$ such that $|f^{(n)}(x)|\leq CA^n n!$ for every $n$ and every $x$ in that neighborhood. The bound must hold on a neighborhood, not only at its center. The <Taylor theorem with Lagrange remainder> proves sufficiency; differentiating a convergent <power series> on a smaller interval proves necessity. A <flat function> shows why pointwise bounds alone are insufficient.
Back to article page
