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Consider the function:

$f(x)= \begin{cases} 1/n \quad &\text{if $x= m/n$ in simplest form} \\ 0 \quad &\text{if $x \in \mathbb{R}\setminus\mathbb{Q}$} \end{cases} $

Prove that the function is continuous at every irrational point and also that the function is not continuous at every rational point. Also, we can say that the function is continuous at some point $k$ if $\displaystyle\lim_{x \to k} f(x)=f(k)$.

I was thinking of doing an epsilon delta proof backwards using the fact that $\mathbb{Q}$ is dense in $\mathbb{R}$ for rational points and irrational points. Any ways on how to expand on this are welcome.

  • 2
    The way to do this is to show that in arbitrarily small neighborhoods of an irrational point $x$, rationals approximating $x$ must have large denominators in reduced form.2012-10-04
  • 1
    Hint: $|\frac ab-\frac xy|=|\frac{ay-bx}{by}|\ge \frac1{by}$ if $\frac ab\ne\frac xy$2012-10-04

2 Answers 2