Does the Taylor series of a real analytic function $f(x)$ converge to $f(x)$ inside the interval of convergence?

Question

Suppose that $f:\mathbb{R}\to\mathbb{R}$ is analytic at $x=0$, and $T(x)$ its Taylor series at $x=0$, with radius of convergence $R>0$. Is it true that $f(x)=T(x)$ whenever $|x|<R$ ?

Answer 1

No. For a counterexample, try

$$f(x) = \begin{cases}e^x & \text{ if } x < 1 \\ 0 & \text{ if } x \geq 1\end{cases}$$

Then $R = \infty,$ but $T(x) = e^x \neq f(x)$ for $x > 1.$

Answer 2

Yes, it is true. It follows from the definition of analytic function that the equality $f(x)=T(x)$ holds on some interval $(-r,r)$. But then you can use the identity theorem: since $(-r,r)$ has accumulation points (actually, each of its points is an accumulation point), and since $(-R,R)$ is connected, $f=T$.