The two iterated integrals are therefore equal. On the other hand, since f xy (x, y) is continuous, the second iterated integral can be performed by first integrating over x and then afterwards over y. But then the iterated integral of f yx − f xy on [ a, b ] × [ c, d ] must vanish. However, if the iterated integral of a continuous function function F vanishes for all rectangles, then F must be identically zero; for otherwise F or − F would be strictly positive at some point and therefore by continuity on a rectangle, which is not possible. Hence f yx − f xy must vanish identically, so that f yx = f xy everywhere.