TITLE: Radii of convergence of real analytic functions
QUESTION [7 upvotes]: I am proposing you a self-posed question, but I do not know whether it makes sense or not. 

Consider an analytic function $f\colon \mathbb R \to \mathbb R$. For every $x$ define $R(x)$ as the radius of convergence of the Taylor series with center in $x$. What can we say about the map $x \mapsto R(x)$? 

For example, for $e^x, \sin{x}, \cos{x}$ we can take $R \equiv +\infty$. Are there any functions for which the map $x \mapsto R(x)$ is constant? What about the continuity? Is there any characterization of the functions $f$ for which $x \mapsto R(x)$ is continuous?
I hope the question is meaningful.

REPLY [5 votes]: Instead of real analytic functions, let us start with some specific class of complex functions which is analytic over some neighborhood of real axis.
There is a theorem by Pólya–Carlson which states that for any power series with
integer coefficients
$$f(z) = \sum_{n=0}^{\infty} c_n z^n\quad\text{ where }\quad c_n \in \mathbb{Z}$$
If it is analytic over the open unit disk, then it is either a rational function or
have the unit circle $S^1$ as its natural boundary of analyticity. 
What this mean is the singularities of $f(z)$ is dense over $S^1$ and one cannot analytic continue $f(z)$ outside the open unit disk.
An example of this is the power series $\displaystyle f(z) = \sum_{n=0}^{\infty} z^{n!}$.
Consider the function 
$$g(z) = f(\frac{e^{-iz}}{2}) = \sum_{n=0}^{\infty} 2^{-n!} e^{-in!z}$$
It is clear $g(z)$ is analytic over $\{z \in \mathbb{C} : \Im z < \log 2 \}$ and has the line $\Im z = \log 2$ as a natural boundary of analyticity. So for every $x \in \mathbb{R}$, the power series expansion of $g(z)$ with respect to $x$ has radius of convergence equal to $\log 2$.
Adding $g(z)$ and $g(-z)$ together, we find the function
$$h(z) = \sum_{n=0}^{\infty} 2^{-n!} \cos( n! z) = \frac12 (g(z) + g(-z))$$
is real over the real axis, complex analytic over the strip $\{ z \in \mathbb{C} : |\Im z| < \log 2 \}$ and has the two lines $\Im z = \pm\log 2$ as natural boundaries.
This give us an example of real analytic function whose radius of convergence is finite and constant over the whole real axis.