Monotone functions of a single variable are well defined, they just keep increasing when the variable increases (or decreases).
I wonder if this concept has a standard generalization to two dimensions.
Obviously, if any function is observed by following an arbitrary path in the XY plane, it can go increasing or decreasing or both. On the other hand, simple functions like affine (a X + b Y + c) or a paraboloid in the first quadrant (f=X^2+Y2, X, Y > 0) are intuitively 2D monotonic.
Is there a way to formalize 2D monotonicity ? Are there monotone surfaces ?
Best Answer
The natural generalization is to monotone functions between partially ordered sets. Now, unlike the case for $\mathbb{R}$, it's debatable whether there is a completely natural partial order on $\mathbb{R}^2$. Instead there are many choices, e.g. lexicographic order (which happens to be a total order) or product order.