Question Number 227438 by Lara2440 last updated on 28/Jan/26
$$\mathrm{1}.\:\mathrm{Let}\:{f}\left({z}\right)=\begin{cases}{\:\mathrm{1}/{q}\:\:\:\:\:{z}={p}/{q}\:,\:{p},{q}\in\mathbb{Z}\:,\:\mathrm{gcd}\left({p},{q}\right)=\mathrm{1}\:,\:{q}>\mathrm{0}}\\{\mathrm{0}\:\:\:\:{z}\in\mathbb{R}\backslash\mathbb{Q}}\end{cases} \\ $$$$\int_{\:\mathbb{R}} \:{f}\left({z}\right)\mathrm{d}{z}=? \\ $$$$\mathrm{2}.\:\mathrm{Show}\:\mathrm{that}\:{f}\left({z}\right)\:\mathrm{is}\:\mathrm{continous}\:\mathrm{function}\:\mathrm{when}\:{z}\in\mathbb{R}\backslash\mathbb{Q} \\ $$$$\mathrm{3}.\mathrm{Show}\:\mathrm{that}\:{f}\left({z}\right)\:\mathrm{is}\:\mathrm{NOT}\:\mathrm{continous}\:\mathrm{when}\:{z}\in\mathbb{Q} \\ $$