For the test of LaTeX typesetting

(It is an easy problem for the sake of testing. )
How to prove f(x)=x\cdot sin(x) is not uniformly continuous?
By sequential criterion all we need to do is to find two sequences (x_n) and (y_n) such that given \epsilon >0, lim|x_n-y_n|=0, |f(x_n)-f(y_n)|\geq \epsilon.
Now we take x_n=2\pi n+\frac{1}{n} and y_n=2\pi n, it can be easily verified that the two sequences will generate the desired results (simply making use of the fact that \lim_{x\to 0} \frac{sin(x)}{x}=1).
Good day;)
Advertisements

Leave a comment

Filed under Mathoholic

Leave a Reply

Fill in your details below or click an icon to log in:

WordPress.com Logo

You are commenting using your WordPress.com account. Log Out /  Change )

Google+ photo

You are commenting using your Google+ account. Log Out /  Change )

Twitter picture

You are commenting using your Twitter account. Log Out /  Change )

Facebook photo

You are commenting using your Facebook account. Log Out /  Change )

Connecting to %s