Reconstructing an analytic function from its Taylor series at a point

Question

Suppose $f:\mathbb{R} \rightarrow \mathbb{R}$ is an analytic function, and at one point $x_1 \in \mathbb{R}$ we know the Taylor series for $f$ centered at $x_1$. Does this determine $f$ uniquely? If not, is there some number of points such that knowing the Taylor series at each point determines $f$?

Answer 1

The question is equivalent to the following: suppose f has an infinite Taylor series expansion around each point in $\mathbb R$ and suppose f is 0 in a neighborhood of some number $a$. Does it follow that $f=0$ everywhere? The answer is YES! If $f(x) \neq 0$ for some $x>a$. Consider $inf \{x>a:f(x) \neq0\}$. If this infimum is $b$ then all the derivatives of f at b are 0 because the functions vanishes in a left-neighborhood of b. The Taylor seried expansion show that f vanishes in a right neighborhood of b too and this contradicts the definition of b. We have proved that $f=0$ on $(a,\infty)$ and a similar argument shows it vanishes in $(-\infty,a)$.