Let F be a holomorphic function defined in an open disk $D(x_0, r)$ where $x_0 ∈ R$. Define a function $f$ in the interval $(x_0 − r, x_0 + r)$ by $f(x) = F(x$) for all $x ∈ (x-0 − r, x_0 + r)$. Suppose that $f$ is a real-valued function on the interval $(x_0 − r, x_0 + r)$. Prove that f is analytic at $x_0$.
I know that since $F$ is holomorphic in the open disk, $F$ has the power series representation that is also analytic in the open disk.
How do I use that to prove the real-valued function is analytic at $x_0$?