Let $f$ be a real analytic function defined on an open interval $I=(a,b)$ with $a,b$ finite.
Is it always possible to extend $f$ to a complex analytic function defined on an open subset $U\subseteq \mathbb{C}$, such that $I\subseteq U$ and there is $\epsilon >0$ such that all open disks of center $x\in I$ and radius $\epsilon$ are included in $U$?
If this is not the case, what about the following apparently weaker statement: Given $f$ as above, for every $\delta>0$ there are a complex analytic extension of $f$ defined on some $U\supseteq (a+\delta,b-\delta)$ and $\epsilon >0$, such that all open disks of center $x\in (a+\delta,b-\delta)$ and radius $\epsilon$ are included in $U$.