Define real-valued function $f$ on $mathbb{R}cap[0,1]$ by setting
$$f(x)=
begin{cases} x,,,text{if $x$ irrational}\
psin(frac{1}{q}),,, text{if $x=frac{p}{q},gcd(p,q)=1$}\
end{cases}
$$
Prove $f$ is continuous at all irrational points of domain. And discontinuous at all rational points.
My try: I want to show when $frac{p}{q}$ close to $xinmathbb{Q}^c$, $q$ will increasing (or inferior increase).