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