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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?

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    Remember that equicontinuity is a condition on a _family_ of functions. If the functions in that family have nothing to do with each other, then you probably won't have equicontinuity. Anyway, if $f_n \to f$ uniformly on a compact set, then $\{f_n\}$ is equicontinuous.2012-07-23

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Simple example: take $I = [0,1]$ and $f_n(x) = x^n$. These are real analytic and even uniformly bounded, but not equicontinuous.

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If you know that the family is bounded and holomorphic on an open region in the complex plane that contains the closure of $I$, then you have equicontinuity by Cauchy estimates on the derivative.

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    @Robert: Than$k$s for spotting the error!2012-07-23