There are questions related to the current question like in this thread or in this thread, but I do have something specific to ask. If this indeed is a duplicate, then please let me know the thread which answers my questions and I will delete this question.
Consider the Taylor series $f(x)$ generated by the infinitely differentiable function $g(x)$ at $x=0$ so that
$$f(x)=g(0)+g'(0)x+\frac12g''(0)x^2+\dots=\sum_{n=0}^\infty\frac{g^{(n)}(0)}{n!}x^n$$
The questions I have are the following.
We do not know if $f(x)$ even converges on any interval. The Taylor series $f(x)$ may not be equal to $g(x)$, its generating function, on any interval, simply because the series may not converge at all. Is that correct?
If we can somehow show, that the power series $f(x)$ converges on an interval $(-R,R)$ for some $R>0$, then it may not be that $f(x)=g(x)$ on that interval as the function $g(x)$ might not be analytic on any interval. Is that correct?
Finally, if we know that $g(x)$ is analytic on an interval $(-R,R)$ for some $R>0$, and that we know that $g(x)$ is continuous at $x=R$, and that the power series $f(x)$ at $x=R$ converges, then $f(x)=g(x)$ at $x=R$. Is that correct?
The idea behind #3 is that as $f(x)$ is convergent at $x=R$, we can apply Abel's theorem which states that
Abel's Theorem: Let $f(x) = \sum_{n=0}^{\infty} a_nx^n$ be a power series that converges at the point $x = R > 0$. Then the series converges uniformly on the interval $[0, R]$. A similar result holds if the series converges at $x = −R$.
to conclude that our function $f(x)$ converges uniformly on $[0,R]$. Then we have the
Continuous Limit Theorem: Let $(f_n)$ be a sequence of functions defined on $A \subseteq \Bbb R$ that converges uniformly on $A$ to a function $f$. If each $f_n$ is continuous at $c \in A$, then $f$ is continuous at $c$.
which implies that our $f(x)$ is continuous at $x=R$. Since $g$ is also given as being continuous at $x=R$, and since we know that $f(x)=g(x)$ on $(-R,R)$ as $f(x)$ is analytic on $(-R,R)$, we have that $$\lim_{x \to R^{-}}f(x) = \lim_{x \to R^{-}}g(x) = g(R)$$.
- Finally, are there any useful theorems to determine if a function is analytic or not?