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I was trying to complete an exercise of a book and tried to solve this question but wasn't able to succeed. I searched it on google but found no results related to this question. Please help.

Andrés Villa
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Ritanshu Singh
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  • For $f(x,y)=\frac{1}{x}+\frac{1}{y}$, we can see that $f(2,2)=1\neq4=2f(1,1)$. This contradicts linearity immediately. – ViHdzP Nov 01 '19 at 04:16
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    $1/x+1/y=0$ is the line $y=-x$ without the point $(0,0)$ – J. W. Tanner Nov 01 '19 at 04:22
  • What does this even mean? When is an equation linear? (It‘s clear that they probably want to hear something along the lines of J.W. Tanner...) – Qi Zhu Nov 01 '19 at 04:24

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If $\dfrac1x+\dfrac1y=0$ then $\dfrac1x=-\dfrac1y$ so $y=-x$, which is of the linear form $y=mx+b$ with $m=-1$ and $b=0$, but $x\ne0$ and $y\ne0$ (because if $z=0$ then $\dfrac1z$ is undefined). So the locus of points $(x,y)$ satisfying $\dfrac1x+\dfrac1y=0$ is a line with a point (the origin) missing.

J. W. Tanner
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