1. Prove in first principle that the following functions are continuous at the origin.
(a) $f(x, y) = \begin{cases} \frac{5x^2y^2}{x^2+y^2} & (x, y) \neq (0,0) \\ 0; & (x, y) = (0,0) \end{cases}$
(b) $g(x, y) = \begin{cases} \frac{xy}{\sqrt{x^2+y^2}} & (x, y) \neq (0,0) \\ 0; & (x, y) = (0,0) \end{cases}$
2. Show that the following functions are discontinuous at the origin.
(a) $f(x, y) = \begin{cases} \frac{x^2-y^2}{x^2+y^2} & (x, y) \neq (0,0) \\ 0; & (x,y) = (0,0) \end{cases}$
(b) $g(x, y) = \begin{cases} \frac{xy^2}{x^2+y^4} & (x, y) \neq (0,0) \\ 2; & (x, y) = (0,0) \end{cases}$
3. A function $f(x, y)$ is defined by
$f(x, y) = \begin{cases} \frac{x^3-y^2}{x^2+y^2} & (x,y)\neq(0,0) \\ 0; & (x, y) = (0,0) \end{cases}$
(a) Find $f_x(x, y)$ and $f_y(x, y)$ when $(x, y) \neq (0,0)$.
(b) Show that $f_x(0, y) = -y$ when $y \neq 0$ and $f_y(x, 0) = x$ when $x \neq 0$.
4. Find the partial derivatives (if they exist) $f_x(0, a)$ when $a \neq 0$, $f_y(b, 0)$ when $b \neq 0$, $f_x(0,0)$ and
$f_y(0,0)$.
$f(x, y) = \begin{cases} \frac{xy^2}{x^2+y^2} & (x, y) \neq (0,0) \\ 0; & (x, y) = (0,0) \end{cases}$
