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Assume that each of {$f_n : [0, 1] \rightarrow R$} is continuously differentiable

I know that if {$f_n'$} is uniformly bounded, {$f_n$} is equicontinuous.

However, the converse is NOT true.

I want to find this example to show that the converse is NOT true

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2 Answers 2

The reason this is not true is because equicontinuity does not restrict the same variation over the whole domain. The problem is essentially the same as showing that a continuous function need not be uniformly continuous if the domain is not compact. A concrete example is $f_n(x) = f(x) = \frac{1}{1-x}$ for all $n$. The family $\{f_n\}$ is equicontinuous on $[0, 1)$ because $f$ is continuous, but $f'$ is not bounded.

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I'm so Sorry Domain is edited Thank you for your comment –  Park Sep 11 '12 at 5:22

Hint: any uniformly convergent sequence of continuous functions on [0,1] is equicontinuous. Think of sine waves of decreasing amplitude and increasing frequency.

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