It is possible for the Taylor series around 0 of a smooth real function $f$ to converge pointwise to $f$ only in a neighborhood $(-r,r)$ of 0 such that $0<r<R$, where $R>0$ is the radius of convergence of the Taylor series?
The case $r=0$ is the only example I can find for "pathological" converging Taylor series, corresponding to flat functions.