Properties of real analytic function [duplicate]

Analytic functions are those that are differentiable infinitely.

let $f$ be a function. My claims are as follows :

C1 : Functions that are analytic are exactly those who have taylor series expansions which converges for all x. (Radius of converges infinite)

C2 : If $f$ is an analytical functions (it's differentiable infinitely) I can create it's taylor series expansion around any point $x=x_0$ such that radius of convergence is infinite.

Are my Claims true? false?

Solution 1:

Hardly anything you wrote is true.

First of all, an analytic function is not the same thing as a function which has derivatives of all orders. It is stronger than that: it is a function $f$ such that, for each $x_0$ in its domain, the Taylor series of $f$ converges to $f$ in some interval $(x_0-\varepsilon,x_0+\varepsilon)$.

Besides, the function $f\colon\mathbb{R}\longrightarrow\mathbb R$ defined by $f(x)=\frac1{1+x^2}$ is analytic, but the radius of convergence of its Taylor series at $0$ is $1$, not $+\infty$. And, of course, this shows that your second claim is also false.