Problem: Let $f: \mathbb{R} ^2 \to \mathbb{R} $ be defined by \[ f(x,y) = \begin{cases} (x^2 + y^2) \sin \left( \frac{1}{x^2 + y^2} \right) , &\text{ if } (x,y)\neq (0,0) ;\\ 0, &\text{ if } (x,y) = (0,0). \end{cases} \] Consider the following statements:

• The partial derivatives $\frac{\partial f}{\partial x} $, $\frac{\partial f}{\partial y} $ exist at $(0,0)$ but are unbounded in any neighbourhood of $(0,0)$.
• $f$ is continuous but not differentiable at $(0,0)$.
• $f$ is not continuous at $(0,0)$.
• $f$ is differentiable at $(0,0)$.

Which of the above statements is/are TRUE?

• I and II only
• I and IV only
• IV only
• III Only