Re: help proving that limit of piecewise function DNE


R. Colacitti wrote:
> R. Colacitti wrote:
> > given:
> >
> > f: R -> R
> >
> > f(x) = { 0 if x is rational
> > { 1 if x is irrational
> >
> >
> > objective:
> >
> > prove that as x->0 , f(x) has no limit (limit does not exist)
> >
> >
> > what I tried and failed miserably with:
> >
> > I noticed that 0 <= f(x) <= 1 for x /in R
> >
> > and since lim 0 != lim 1 as x->0 , I claimed that lim f(x) as x->0 does
> > not exist
> >
> > *BASED ON A COROLLARY OF THE SQUEEZE THEOREM (something which isn't
> > generally true)
> >
>
> I should have said "CONVERSE" not "COROLLARY"
>
> >
> > Any hints on how I should start in trying to prove that the limit does
> > no exist?

As a hint, every neighborhood of 0 contains an irrational number.

Mike

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