I have a problem understanding the idea behind Analytic functions. (Please correct me on my terminologies while I state my problem).
An analytic function, is a function that has a power series that converges relative to some point $a$. I know that some functions like $f(x)=x$ are analytic no matter which point $x$ is chosen in the domain. The function $f(x)$ is always equal to its Taylor series expansion.
For certain functions, I know that the $x$ you pick must be very specific, relative to a point $a$ you're taking within a 'radius of convergence'. Otherwise the series expansion diverges.
1) When are functions considered 'analytic'? My problem lies in the fact that 'analytic' is referring to the entire function, and yet by definition it relies on the $x$ you pick, and the $a$ you pick. So for the case when there is a radius of convergence, does that mean that a function $f(x)$ is analytic if and only if $x$ is within the radius of convergence?
2) There is a third case when dealing with the convergence of the power series. When $x=a$, the power series trivially converges to 0. Yet, does that mean since it does converge, the particular $f(x)$ taken must be analytic at $x$?