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The Equality of the Cross Difference Ratios for a Binary Function

A fundamental theorem concerning partial derivatives is that for a binary function f(x, y), providing that the
derivatives exist,

∂/∂y(∂f/∂x) = ∂/∂x(∂f/∂)
which is usually expressed as
∂²/∂y∂x = ∂²/∂x∂y

The purpose of this article is to show that the analogous theorem concerning difference ratios also holds true.
Let f(x, y) be a binary function. A difference ratio at (x, y) could be [f(x+h, y)−f(x−h, y)]/(2h).
One cross difference ratio centered on (x, y) is