GATE XE 2023 | Question: 2

Question

Let $f: \mathbb{R}^{2} \rightarrow \mathbb{R}$ be a function defined by 

$f(x, y)=\left\{\begin{array}{cc}\frac{x y}{|x|+y}, & y \neq-|x| \\ 0, & \text { otherwise. }\end{array}\right.$

Then which one of the following statement is TRUE?

• A. $f$ is NOT continuous at $(0,0)$.
• B. $\frac{\partial f}{\partial x}(0,0)=0$, and $\frac{\partial f}{\partial y}(0,0)=1$
• C. $\frac{\partial f}{\partial x}(0,0)=1$, and $\frac{\partial f}{\partial y}(0,0)=0$
• D. $\frac{\partial f}{\partial x}(0,0)=1$, and $\frac{\partial f}{\partial y}(0,0)=1$