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Can a 2-dimensional scalar field have a discontinuous contour curve? How about contour curves that intersect -- possible?

On a related note: can a vector field have a domain that is not defined over a continuous region??

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    Then they should have specified which scalar functions they were talking about. ;-) Anyway, physicists may very well be interested in discontinuous scalar fields, for example an idealized shock wave in a fluid, where there's a jump in the pressure.2012-09-30

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Yes. For example, the scalar field $f (x,y) := x y$ has discontinuous contour curves. Note that $f (x,y) = 1$ yields two hyperbolas:

Two hyperbolas

Animated plot courtesy of Wikipedia.

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Yes, there are many ugly functions.. it also depends what do you mean by a 2 dimensional scalar field (because one usually consider them at least continuous by definition).

So, for example, take the following function: $(x,y) \mapsto \left\{ \begin{matrix} 0 & \text{if }x,y\in\mathbb Q \\ 1 & \text{else} \end{matrix} \right. $

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    "it also depends what do you mean by a 2 dimensional scalar field (because one usually consider them at least continuous by definition)." This is precisely what I'm trying to figure out: are scalar fields **continuous** functions by definition?? My intuition tells me yes (and also, the concept of contour curves, etc., all seem to point towards this), yet I can't find any definition that mandates that scalar fields be **continuous** functions.2012-09-30