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Let $u$ be a non-constant harmonic function on $\mathbb{R}$. Show that $u^{-1}(c)$ is unbounded.

I am not getting what theorem or result to apply. Could anyone help me?

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    Are you sure there is not something missing here? The harmonic functions on $\mathbb{R}$ are just the linear functions, and then the result is definitely false.2012-06-24

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Let $u(x)=x$, for all $x \in \mathbb{R}$. Then $u''(x)=0$ for all $x$. But $u^{-1}(\{c\}) = \{c\}$. I think somebody is cheating you :-)

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    @TaxiDriver If you still want to know the answer for n>1 (which is quite a different story from $n=1$), you should post another question with a link to this one.2013-06-15