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In an attempt to better understand the definition of an equicontinuous family of continuous functions, I want to find a simple non-example.

My intuition says that the family $\{f_n\colon[0,1]\to\mathbb R\}_{n\in\mathbb N}$ given by $f_n(x)=x^n$ is not equicontinuous, but I do not know how to show this.

Any help is appreciated.

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    Can you elaborate on your "intuition?" Where do you think this family family fails to be equicontinuous?2012-07-06
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    Could this also be tagged functional-analysis?2012-07-06
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    @MattN. In my opinion, not really. While the concept of equicontinuity plays a role in a functional analysis, the question here asks for the explicit construction of a counterexample, giving the question a much more real-analytic flavor.2012-07-06
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    @PZZ Ok. Thanks for the comment!2012-07-06

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