Another Proof of Clairaut's Theorem

This note gives an alternate proof of Clairaut’s theorem—that the partial derivatives of a smooth function commute—using the Stone–Weierstrass theorem. Most calculus students have probably encountered Clairaut’s theorem. Theorem. Suppose that f : [a, b] × [c, d] → R has continuous second-order partial derivatives. Then fxy = fyx on (a, b)× (c, d). The proof found in many calculus textbooks (e.g., [2, p. A46]) is a reasonably straightforward application of the mean value theorem. More sophisticated techniques—Fubini’s theorem and Green’s theorem—can each be used to give easy proofs (for instance, [1, p. 61], exercise 3-28). The proof here relies on the density of two-variable polynomials in C([a, b] × [c, d]). More precisely, we use the following version of the Stone–Weierstrass theorem. Theorem. Let g ∈ C([a, b] × [c, d]). There is a sequence pn(x, y) of two-variable polynomials such that pn → g uniformly. Applying the theorem to the continuous function fxy gives a sequence of polynomials pn such that |pn(x, y)− fxy(x, y)| < (n) for all (x, y) ∈ [a, b] × [b, c] where limn→∞ (n) = 0. Therefore, for any rectangle R = [x1, x2] × [y1, y2] ⊂ [a, b] × [c, d], ∣∣∣∣∫∫ R pn dxdy − ∫∫ R fxy dxdy ∣∣∣∣ < (n)A(R), (1) where A(R) = (x2 − x1)(y2 − y1) is the area of the rectangle R. Observe that ∫∫