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Question Number 227438 by Lara2440 last updated on 28/Jan/26

1. Let f(z)= { (( 1/q z=p/q , p,q∈Z , gcd(p,q)=1 , q>0)),((0 z∈R\Q)) :} ∫_( R) f(z)dz=? 2. Show that f(z) is continous function when z∈R\Q 3.Show that f(z) is NOT continous when z∈Q

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

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