# Prove continuity

### Prove continuity

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.
seonguvai

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### Re: Prove continuity

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

Hope this helped,

R. Baber.
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