MathDB

Let

f: [0,1]^2\to\mathbb{R}

be a continuous function on the closed unit square such that

\frac{\partial f}{\partial x}

and

\frac{\partial f}{\partial y}

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

(x_0,y_0)

(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.

Problem Statement