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Lets $f:\mathbb{R}^2 \to \mathbb{R} $, where $f$ is harmonic, continuous and non-constant. How do I go about showing that the level curves of $f$ are smooth?

Thanks!

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    How are the level curves of a function $f:\mathbb{R}^2 \to \mathbb{R}^2$ defined? If you mean the set of arguments where the function takes on a specific value, this will typically be a point, not a curve. Did you mean $f:\mathbb{R}^2 \to \mathbb{R}$?2011-09-30
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    Even for $f:\mathbb{R}^2 \to \mathbb{R}$ the level curves may not be smooth. For instance, $f(x,y)=y-|x|$.2011-09-30
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    Yes sorry I meant, $f:\mathbb{R}^2 \to \mathbb{R}$.2011-09-30
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    Note: lhf's comment was made when the requirement that $f$ is harmonic was an option in parentheses. If $f$ is harmonic, it is smooth. In particular, that means that the requirement of continuity is now redundant, since harmonic functions are necessarily continuous.2011-09-30

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