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Anyone, who can help me to decide whether $v(x,y)=x\sqrt{x+y}+xy+7$ is a injective function. It is easy for me to decide whether a function of one variable is injective, but I am having trouble with more variable cases like this one.

Hope someone can help

Michael Hardy
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frankfranfrank
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    $v(0,y)=7;;\forall y$ – lulu Feb 11 '16 at 17:53
  • Consider the level sets of this function (the sets $C_r={(x,y)\mid v(x,y)=r}$). $v$ is constant on each such (nonempty) set. If you can find a level set containing more than one point, you will have shown the function is not injective. – MPW Feb 11 '16 at 17:55
  • An injective function from $\Bbb R^2$ to $\Bbb R$ is not impossible, but its expression will not be so simple... – ajotatxe Feb 11 '16 at 17:56
  • @ajotatxe An injective function from $\mathbb R^2 $ to $ \mathbb R $ with $x$ or $y$ continuous is impossible. See https://math.stackexchange.com/questions/1695302/does-there-exist-an-injective-function-f-mathbb-r2-to-mathbb-r-such-that?rq=1 – john Nov 06 '19 at 07:53

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If you move in a direction in the $(x,y)$-plane that is perpendicular to the gradient, you're following a level curve of this function. If there are non-empty level curves, then clearly the function is not one-to-one.

Michael Hardy
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