We classify, under reasonable assumptions, all differentiable functions f for which the ‘secant method’ [xf(y) − yf(x)]/[f(y) − f(x)] is continuous and associative. Further, we classify all differentiable functions for which the similar type of addition xf(y) + yf(x) is associative.
Northshield, S. (2009). On two types of exotic addition. Aequationes Mathematicae, 77, 1-23.