A Question on Uniform Convergence

**garunas** · January 11th 2011, 12:09 PM

Give an example of a uniformly convergent sequence $(f_{n})_{n\in\mathbb N}$ of continuously differentiable functions such that $(f\prime_{n})_{n\in\mathbb N}$ is not uniformly continuous.

Just preparing for my exams in a few weeks. Need to know the answer to this one to have as an example. Just in case!

**FernandoRevilla** · January 11th 2011, 12:26 PM

Choose:

$f_n(x)=\dfrac{\sin nx}{n}$

Fernando Revilla

**garunas** · January 11th 2011, 12:28 PM

Many many thanks. Simple when you know how isn't it??

**FernandoRevilla** · January 11th 2011, 12:32 PM

Originally Posted by garunas

Many many thanks. Simple when you know how isn't it??

No wonder. It is a typical counter example.

Fernando Revilla

**emakarov** · January 11th 2011, 12:39 PM

Originally Posted by garunas

Give an example of a uniformly convergent sequence $(f_{n})_{n\in\mathbb N}$ of continuously differentiable functions such that $(f\prime_{n})_{n\in\mathbb N}$ is not uniformly continuous.

Sorry, what exactly is not uniformly continuous? Should $f'_n$ be not uniformly continuous for every $n$ ?

**FernandoRevilla** · January 11th 2011, 12:46 PM

Originally Posted by emakarov

Sorry, what exactly is not uniformly continuous? Should $f'_n$ be not uniformly continuous for every $n$ ?

I supposed that is a typo and he meant " ... $(f'_n)$ not uniformly convergent".

Fernando Revilla

**garunas** · January 11th 2011, 01:30 PM

Yeah sorry I am a bit new to Latex

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