In my answer to this question on MU, I suggested that the OP think about the difference between real-differentiable and complex-differentiable functions by using a sort of finitary analogue. One way to make this precise is the following. The finitary analogue of a real-differentiable function is just a function $f : \mathbb{Z} \to \mathbb{Z}$, whose finitary derivative is the first difference $f(z + 1) - f(z)$. The first difference can be arbitrarily nasty; in particular it does not have to grow in any kind of smooth way, which is a reasonable analogue of what happens for arbitrary real-differentiable functions.

The finitary analogue of (the real part of) a complex-differentiable function, on the other hand, is a discrete harmonic function $f : \mathbb{Z}^2 \to \mathbb{Z}$ e.g. one which satisfies

$$\frac{f(x+1, y) + f(x-1, y) + f(x, y+1) + f(x, y-1)}{4} = f(x, y).$$

Such functions are clearly more constrained than arbitrary functions $\mathbb{Z} \to \mathbb{Z}$. Is there a reasonable finitary statement about the properties of such functions which is analogous to the statement that harmonic functions are smooth?

(The best case would be if one could deduce the infinitary statement from the finitary one, but maybe this is difficult. Also, I can't think of good tags for this question, so feel free to retag.)

reallywant to consider integer-valued functions only? – Mariano Suárez-Alvarez♦ Jul 28 '10 at 3:48