(adsbygoogle = window.adsbygoogle || []).push({}); 1. The problem statement, all variables and given/known data

f(x) = {x^2 [tex]x \in Q[/tex]

-x^2 [tex]x \in R/Q [/tex]

At what points is f continuous?

2. Relevant equations

continuity: for every [tex]\epsilon > 0 [/tex] there exists [tex]\delta > 0 d(f(x),f(p)) < \epsilon[/tex] for all points [tex]x\inE[/tex] for which d(x,p) < [tex]\delta[/tex]

3. The attempt at a solution

Alright my initial thought was that it would not be continuous at any point in Q, because for any two rationals there is an irrational between them (this is correct?), but then it would be continuous at all irrationals from a theorem (4.6 in Rudin) for [tex]p\in Q[/tex], lim(x-> p) f(x) = -p^2 [tex]\neq[/tex] p^2 = f(p)

However, then this function is continuous at irrationals. For [tex]p\in R/Q[/tex], lim(x-> p) f(x) = -p^2 = f(p)

is this reasoning sound ok?

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# Continuity of x^2 and -x^2

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