Question 2

a. Define ε–δ definition of continuity of a function of two variables $f : D \subset \mathbb{R}^2 \to \mathbb{R}$. Using ε–δ definition, discuss the continuity of the function$$f(x,y) = \begin{cases} \dfrac{x^2+y^2}{3+\sin x}, & \text{if } (x,y) \neq (0,0) \\ 0, & \text{if } (x,y) = (0,0) \end{cases} \quad \text{at } (0,0).$$ (2 marks)

b. Define the differentiability of function of two variables $f : D \subset \mathbb{R}^2 \to \mathbb{R}$ at the point $(x,y)$. Check the differentiability of the function$$f(x,y) = \begin{cases} (x^2+y^2)\sin\left(\dfrac{1}{x^2+y^2}\right), & \text{if } (x,y) \neq (0,0) \\ 0, & \text{if } (x,y) = (0,0) \end{cases} \quad \text{at } (0,0).$$ (2 marks)