Chapter 14.4, Problem 47E

To determine

(a)

Prove that $g\left(x\right)$ is continuous at point $\left(0,0\right)$, use the polar coordinates.

To determine

(b)

Prove that $g_x\left(0,0\right)\text{and}{g}_y\left(0,0\right)$ exists and equal to 0, use the limit definitions.

To determine

(c)

Prove that the linearization of $g\left(x,y\right)$ at $\left(0,0\right)$ is $L\left(x,y\right)=0$.

To determine

(d)

Prove that $g\left(x,y\right)$ is not differentiable at $\left(0,0\right)$.