TITLE: What exactly is an "analytic function"?
QUESTION [7 upvotes]: This is for real analysis so I'm not worried about complex analytic functions. 
The definition in my book just says: "A function f(x) which is represented by a power series with a positive radius of convergence is said to be 'real analytic at the origin', or simply 'analytic'."
To me it seems like this definition ends prematurely. Like when I was reading it I expected there to be an "if" and then a list of qualifications. 
I guess I just don't really understand the rationale behind creating this definition in the first place. What would even be the point of representing a function by a power series that didn't have a positive radius of convergence? If the power series doesn't converge for any values of x, it's not really representative of anything, right?
I feel like this is actually obvious and I'm just overthinking it, but is this definition literally just giving a name to functions that can be represented as power series that actually converge for x values in some radius? There was an exercise earlier in the chapter before analytic functions were defined that asked us to find an infinite power series which represented a function; was that function analytic as well?
Thanks in advance and sorry if this is a silly question.

REPLY [3 votes]: Why do we define any classification? So we can develop a language that can quickly say whether or not a structure exhibits a certain property.
Some functions can't be represented by a power series while others can. We have this definition to put functions into this analytic or not analytic category -- we don't have to say "$f$ can be expanded into a power series", we just have to say "$f$ is analytic." Another thing, a lot of the time we can have equivalent definitions of the same thing. This way we can avoid confusion by calling it by one name.
Of course, we make classifications only when they are useful. In this case, we can get a lot of very useful properties from the fact that a function is analytic: for example, smoothness follows from analyticity.