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Let $D_n$ be the open disc of radius $n$ with centre at the point $(n,0)\in\mathbb{R}^2$. Does there exist a function $f:\mathbb{R}^2\to\mathbb{R}$ of the form $f(x,y)=ax+by$ such that $$\cup_{n=1}^\infty D_n=\{(x,y)\mid f(x,y)>0\}\;?$$ If your answer is 'Yes', give the values of $a$ and $b$.

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The given set is open set, I think a function $f(x,y)=x$ or $y$ can do the job, but I am not sure, I also want to know whether inside the question any deep result is hidden from analysis.

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1 Answer 1

up vote 6 down vote accepted

Hint: Note that $\cup_{n=1}^\infty D_n=\{(x,y)\in\mathbb{R}^2\mid x>0\}$.

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