1) Let $f_n:I \to \mathbb{R}$, $n \in \mathbb{N}$, be a family of functions on an Interval $I \subset \mathbb{R}$. If all functions are real analytic, i.e. identical to their Taylor series expansion near every point of $I$, does this imply that the family is equicontinuous? One may assume $I$ to be bounded or even compact here.
2) If not, do you know any useful theorems under which additional assumptions this could be true?