0x67320866

Let \(f\) be smooth function. If the Taylor series converges and the rest in the Taylors formula vanishes at every point, the function itself is analytic .

This follows by

\begin{equation*} f(x)=f(x-\xi+\xi)=\sum_{\lvert \alpha\rvert\le k} \frac{D^{\alpha}f(x-\xi)}{\alpha!}\xi^\alpha + R_k(x-\xi,\xi). \end{equation*}