Does there exist an infinitely differentiable function $f:U\to\mathbb{R}$, where $U$ is open subset of $\mathbb{R}$, such that
- the Taylor series of $f$ at $x=x_0\in U$ has radius of convergence $R>0$
- $f$ equals its Taylor series only on the subinterval $(x_0-r,x_0+r)$, where $\color{red}{0<}r<R$
The customary examples of smooth real functions that fail to be analytic, e.g. $e^{-1/x}$ or $e^{-1/x^2}$ at $x=0$, have $R=\infty$ but $r=0$. The substance of the question is whether we can find a less extreme example for which analyticity at $x=x_0$ gives out only at a nonzero radius smaller than the radius of convergence of the Taylor series.
Note: I don't really know complex analysis, but I know that the easiest path to whatever the truth is here is probably through the complex domain.