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

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

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Where are the two sequences x_n and y_n separately supposed to converge?
 
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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
 
1,631
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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

[tex]x_n=\frac{1}{n}[/tex] [tex]y_n=\frac{1}{n^2}[/tex] these both converge to zero and also

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

Also:

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

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