I'm working on a problem in game theory, and I've run into a proposition that I don't know how to prove. Leaving out the game theory bit, I have a family of functions defined over a k-dimensional simplex onto the real numbers. Each function is fairly well-behaved: continuous and differentiable in most places, possibly including a small number of discontinuities.
Based on simulations, I strongly suspect that with measure one, a function within this class has a unique global maxima. How do I prove it?