Exercise 19

Use the properties of continuity to determine the set of points at which each of the following functions is continuous. Justify your answers.

1. The function $f$ defined by $f(x,y) = \frac{x+2y}{x-y}$

2. The function $g$ defined by $g(x,y) = \frac{\sin(x)}{1+e^y}$

3. The function $h$ defined by $$\begin{aligned} h(x,y) = \begin{cases} \frac{xy}{x^2+y^2} & \text{ if } (x,y) \neq (0,0) \\ 0 & \text{ if } (x,y) = (0,0) \end{cases} \end{aligned}$$

4. The function $k$ defined by $$\begin{aligned} k(x,y) = \begin{cases} \frac{x^2y^4}{x^2+y^2} & \text{ if } (x,y) \neq (0,0) \\ 0 & \text{ if } (x,y) = (0,0) \end{cases} \end{aligned}$$