A Question on Uniform Convergence

• January 11th 2011, 12:09 PM
garunas
A Question on Uniform Convergence
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!
• January 11th 2011, 12:26 PM
FernandoRevilla
Choose:

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

Fernando Revilla
• January 11th 2011, 12:28 PM
garunas
Many many thanks. Simple when you know how isn't it??
• January 11th 2011, 12:32 PM
FernandoRevilla
Quote:

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
• January 11th 2011, 12:39 PM
emakarov
Quote:

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 $n$?
• January 11th 2011, 12:46 PM
FernandoRevilla
Quote:

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
• January 11th 2011, 01:30 PM
garunas
Yeah sorry I am a bit new to Latex