Your question is a bit unclear. F.i., the solution V(r) of the Laplace or Poisson equations may always be considered to define equipotential surfaces, defined by the set of points where V(r) has a fixed given value. 

How complicated such surfaces may be (as in the problem of diffusion limited aggregation) is something else. 

Spiros> cut the plane

You mean that f.i. the line of points satisfying Re f(z) = c, for some real number c, is cutting the plane? Isn't that the equivalent of a surface in two dimensions?

Or, are you searching for closed surfaces? Earnshaws theorem tells us that a solution of the Laplace equation cannot have maxima or minima, that must have some implications.

The question, as posed, can't be answered, since there isn't any logical relation between the statement about an equation, in general,  and the statement about an equipotential surface: the two are completely distinct notions. 

It's not true that the solutions of the Laplace (or the Schrödinger) equation always form loops, or knots-that depends on the boundary conditions. The physics of the Laplace equation is completely different from that of the Schrödinger equation, incidentally, so it's not useful to mention them together. And it's not true that the solutions can't jump in value: in the presence of electric charges, for instance, Gauss' law implies that solutions of the Laplace equation  can't be loops-the electric field lines  end on the surfaces the charges are localized on and the electric field is discontinuous there. 

Of course it's possible-however the discussion is meaningless unless further details are specified. So it's more useful to discuss a specific problem. 

Dear Spiros, from your answer `` I believe the answer to my question is that every continuous function of space has closed surfaces of constant values. '' it seems to me you may be interested in the properties of the surfaces of constant V(x) for an arbitrary continuous scalar function V(x) with a vector argument x. It is then true that these surfaces are all disjoint, in other words, they do not meet, and most of them are smooth. They need not be bounded. Consider the following example in R^2

H(x_1,x_2) = x_1^2/2 + cos(x_2)

You will see that the curves H(x_1,x_2)=E are closed (egglike) when E1. When E=1, the curve is not continuous. According to Sard's theorem, this last phenomenon can only occur for a set of values of E which has measure zero, so it is, in a sense, uncommon.

My remarks do not, of course, have any bearing on the Laplace or Schroedinger equation. I am just talking in general terms of the properties of surfaces in which a given differentiable function is constant.

Sorry, a big mistake in my previous message: I said ``for E=1 the curve is not continuous''. I meant, of course, the curve is not fifferentiable. It has ``corners''. Another point: Sard's theorem is only true for sufficiently differentiable functions.