Measure, Integration & Real Analysis · Problem 5C.14

Banach Spaces — Sheldon Axler

Mathematics — Measure Theory · mixed-partial-derivatives-counterexample

Problem

Define

f : \mathbf{R}^2 \to \mathbf{R}

f(x, y) = \begin{cases} \dfrac{xy(x^2 - y^2)}{x^2 + y^2} & \text{if } (x, y) \neq (0, 0), \\ 0 & \text{if } (x, y) = (0, 0). \end{cases}

(a) Prove that

D_1(D_2 f)

and

D_2(D_1 f)

exist everywhere on

\mathbf{R}^2

. (b) Show that

\bigl(D_1(D_2 f)\bigr)(0, 0) \neq \bigl(D_2(D_1 f)\bigr)(0, 0)

. (c) Explain why (b) does not violate 5.48.