- #1
billy2908
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Homework Statement
Let f(x)=[tex]x^{1/3}[/tex] show that it is uniform continuous on the Real metric space.
Homework Equations
By def. of uniform continuity [tex]\forall\epsilon[/tex]>0 [tex]\exists\delta>0[/tex] s.t for [tex]\forall x,y\in\Re[/tex] where |x-y|<[tex]\delta[/tex] implies |f(x)-f(y)|< [tex]\epsilon[/tex]
The Attempt at a Solution
I started w/ |[tex]x^{1/3}[/tex] -[tex]y^{1/3}[/tex]|* (|[tex]x^{2/3}[/tex] +[tex]xy^{1/3}[/tex]+[tex]y^{2/3}[/tex]|/|[tex]x^{2/3}[/tex] +[tex]xy^{1/3}[/tex]+[tex]y^{2/3}[/tex]|)
=|x-y|/(|[tex]x^{2/3}[/tex] +[tex]xy^{1/3}[/tex]+[tex]y^{2/3}[/tex]|)
But it doesn't seem to be uniform cont. if I set [tex]\delta[/tex]=[tex]\epsilon[/tex]*(|[tex]x^{2/3}[/tex] +[tex]xy^{1/3}[/tex]+[tex]y^{2/3}[/tex]|)