Question

Let f(x)= cos 1/x is not uniformly continuous on (0, infinity)

Answer 1

True. For $x_n=\frac{1}{2\pi n},y_n=\frac{1}{\frac{\pi}{2}+2\pi n}$

$f\left( x_n \right) -f\left( y_n \right) =\cos \left( 2\pi n \right) -\cos \left( \frac{\pi}{2}+2\pi n \right) =1-0=1$

Meanwhile $\left| x_n-y_n \right|=\left| \frac{1}{2\pi n}-\frac{1}{\frac{\pi}{2}+2\pi n} \right|=\frac{1}{2\pi n\left( n+4 \right)}\rightarrow 0,n\rightarrow \infty$

This contradicts with the definition of uniform continuity.