List of top Real Analysis Questions on Continuity and Differentiability in Several Variables asked in GATE MA

Let \(f: \mathbb{R}^2 \to \mathbb{R}\) be a function defined as follows:
\[ f(x,y) = \begin{cases} \dfrac{x\sqrt{x^2+y^2}}{|x|} & \text{for } x \neq 0 \\ 0 & \text{for } x = 0 \end{cases} \]
Which of the following statements is/are TRUE?