Analytic versus $C^{\infty}$ example

Question

I am currently reading Loring introduction to differential manifolds. There is something I don't really understand in the example below.

So a function $f$ is real analytic at $p$ if in some neighborhood of $p$ it is equal to its Taylor series at $p$. In the example below, we can see f is analytic right? I mean if we get the Taylor expression of f and evaluate at p we find that the Taylor series of this function is zero, which has the same value as f at 0?

Answer 1

The key point is that a function $f$ is real analytic at $p$ if in some neighborhood of $p$ it is equal to its Taylor series at $p$. In your example, no such neighborhood exists at $p=0$ because $f(x)>0$ for any $x>0$.

There is a detailed article in Wikipedia regarding your example:

https://en.wikipedia.org/wiki/Non-analytic_smooth_function

Read in particular the section:

https://en.wikipedia.org/wiki/Non-analytic_smooth_function#The_function_is_not_analytic

Answer 2

For $f(x)$ to be analytic around $x=0$ we would need its Taylor series to coincide with $f(x)$ in some neighborhood of $x=0$, but this is not the case since the Taylor series of $f(x)$ is the zero series, and $f(x)$ is strictly positive to the right of $x=0$.