Question

Let 𝑓:ℝ×ℝ→ℝ be a function defined by
$f(x,y)=\begin{cases} \frac{xy(x+y}{x^2+y^2} & \quad \text{if }(x,y)≠(0,0),\\\ 0, & \quad if\,(x,y)=(0,0). \end{cases}$
Then, which of the following statements is/are TRUE?

• A. 𝑓 is continuous on ℝ × ℝ
• B. The partial derivative of 𝑓 with respect to 𝑦 exists at (0, 0), and is 0
• C. The partial derivative of 𝑓 with respect to 𝑥 is continuous on ℝ × ℝ
• D. 𝑓 is NOT differentiable at (0, 0)

Answer 1

Answer: (A), (B), (D)

𝑓 is continuous on ℝ × ℝ

Answer 2

The correct options are: A, B and D