Question 1

a. Test the function $f(x)$ for continuity at the point $x = 0$, given $f(x) = x\sin(1/x),\ x \neq 0$, and $f(0) = 0$ at $x = 0$. (3 marks)

b. Examine for continuity and differentiability of the function$$f(x,y) = \frac{xy^2}{x^2+y^2},\ \text{when } (x,y) \neq (0,0), \quad f(0,0) = 0 \text{ at } (x,y) = (0,0).$$ (5 marks)