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I'm having a little bit of difficulty with this question. Could someone give some advice?
Consider the function:
f(x) = 1/b if x is rational and x = a/b in lowest terms and b>0
and f(x) = 0 if x is irrational.
Show that f(x) is continuous exactly at irrational points.
My thoughts so far: If a/b is in lowest terms, I'm guessing it means that and a and b are relatively prime? So b would equal a/x, when x is a rational number? I'm then getting that f(x) = x/a, when x is a rational number, but why would this not be continuous for any intervals? Shouldn't the interval of a from (0,1] work at least?
Any help would be much appreciated!
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I was surprised to see continuity at irrational point because I had always thought irrational space isolated and discontinuous but on thinking it looks otherwise.e.g
{1}Vasudhaiva Kutumakam.{The whole Universe is a family.}
(2)Yatra naaryasthu poojyanthe Ramanthe tatra Devataha
{Gods rejoice at those places where ladies are respected.}
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Thanks Thickhead!
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