Consider the function $f:[0,1]\rightarrow\mathbb{R}$, where

• $f$ is real-analytic on the open interval $(0,1)$
• $f$ is bounded on the closed interval $[0,1]$ (ie. there is some constant $C$ such that $-C\leq f(x)\leq C$ for $x\in[0,1]$).

Is it true that there is a real-analytic continuation of $f$ to the interval $[-\epsilon, 1+\epsilon]$ for some small positive $\epsilon$? If not, what conditions can be added to make it true?

Suggestions for books (or other references) where I could have learned to answer this myself would also be appreciated.