Prove continuity

Prove continuity

Postby seonguvai » Wed Apr 08, 2015 3:50 am

7. BONUS (4 points) Let f be the function defined by:

. . . . .f(x)={x2−x2 if x is rational if x is irrational

Is f continuous at x = 0? If so, prove it. If not, prove that it is not.
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Re: Prove continuity

Postby Guest » Sat Apr 11, 2015 3:06 am

Using Weierstrass' definition of continuity it is easy to see that [tex]f(x) = x^2[/tex] or [tex]-x^2[/tex] is continuous at [tex]x=0[/tex].
http://en.wikipedia.org/wiki/Continuous ... _functions
Under this definition all we are required to do to show continuity at [tex]x=0[/tex] is find a function [tex]\delta(\epsilon)>0[/tex] (for [tex]\epsilon>0[/tex]) such that [tex]|x|<\delta(\epsilon)[/tex] implies [tex]x^2<\epsilon[/tex] (we know that [tex]|f(x)-f(0)|=x^2[/tex]). One such function is [tex]\delta(\epsilon) = \sqrt(\epsilon)[/tex] (it is easy to check that this works).

Hope this helped,

R. Baber.
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