I'm trying to show a function has non-uniform continuity

[rbzima]

I'm trying to show a function has non-uniform continuity, and I can't seem to think of 2 sequences (x_n) and (y_n) where |(x_n) - (y_n)| approaches zero, where f(x) = x³. Can anyone think of two sequences?

 

[sutupidmath]

Where are the two sequences x_n and y_n separately supposed to converge?

 

[rbzima]

Where are the two sequences x_n and y_n separately supposed to converge?

|x_n - y_n| need to converge to 0.
|f(x_n ) - f(y_n)| needs to converge to some real number, where f(x) = x^3

 

[sutupidmath]

I don't know if i am getting you right, but if there are no additional conditions imposed there, then i think the following would work

$$x_n=\frac{1}{n}$$ $$y_n=\frac{1}{n^2}$$ these both converge to zero and also

let $$|a_n=x_n-y_n=\frac{n-1}{n^{2}}|-->0, \ \ as \ \ n-->\infty$$

Also:

$$|f(x_n)-f(y_n)=\frac{1}{n^{3}}-\frac{1}{n^6}=\frac{n^3-1}{n^6}|-->0, \ \ as \ \ \ n-->\infty$$