prove that $f$ is continuous over $[0,1]\cap\mathbf{Q}^{c}$

Question

Consider the function $f:[0,1]\to\mathbf{R}$, with

$$f(x)=\begin{cases}\frac{1}{b} ,\quad x=\frac{a}{b}\in[0,1]\cap\mathbf{Q}, \ (a,b)=1 \\ 0 , \quad x \in [0,1]\cap\mathbf{Q}^{c} \text{ or } x=0 \end{cases}$$

Prove that $f$ is continuous for every $r\in [0,1]\cap\mathbf{Q}^{c}$

Could I have some kind of hint or help?

Possibly related post http://math.stackexchange.com/questions/207118/prove-continuity-on-a-function-at-every-irrational-point-and-discontinuity-at-ev — ForgotALot, Mar 13 '16 at 16:03
Use the fact that if a sequence of rationals converges to an irrational number, then the sequence of denominators of the sequence diverges to infinity. — DonAntonio, Mar 13 '16 at 16:08
@Joanpemo ok here you go http://math.stackexchange.com/questions/1697891/prove-that-sequence-of-rationals-goes-to-infinity — karhas, Mar 14 '16 at 23:44

score 3 · Answer 1 · answered Mar 13 '16 at 16:05

If $\varepsilon > 0$ and $x$ is irrational, take an $N \in \mathbb N$ with $1/N < \varepsilon$. There are only finitely many rationals $p/q \in[x-1, x+1]$ where $(p,q) = 1$ and $q \le N$. Since none of these are equal to $x$, take $\delta$ small enough so that $(x-\delta, x+\delta)$ does not contain of these rationals. This $\delta$ will prove continuity.

score 1 · Accepted Answer · answered Mar 13 '16 at 16:07

1

Choose an irrational $p\in[0,1]$, given any $\epsilon>0$, let $N:=\lceil1/\epsilon\rceil$. Enumerate every $b/a\in\Bbb Q\cap[0,1],\,(a,b)=1,\,a\le N$ as $q_1,\cdots,q_m$. Now choose $$\delta:=\min_{1\le k\le m}|p-q_k|. $$

answered Mar 13 '16 at 16:07

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prove that $f$ is continuous over $[0,1]\cap\mathbf{Q}^{c}$

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