Let f:[0,1]2→R be a continuous function on the closed unit square such that ∂x∂f and ∂y∂f exist and are continuous on the interior of (0,1)2. Let a\equal{}\int_0^1f(0,y)\,dy,\ b\equal{}\int_0^1f(1,y)\,dy,\ c\equal{}\int_0^1f(x,0)\,dx and d\equal{}\int_0^1f(x,1)\,dx. Prove or disprove: There must be a point (x0,y0) in (0,1)2 such that
\frac{\partial f}{\partial x}(x_0,y_0)\equal{}b\minus{}a and \frac{\partial f}{\partial y}(x_0,y_0)\equal{}d\minus{}c. Putnamfunctionintegrationvectorcalculusderivativeanalytic geometry