I need to prove the following formula:
$x^a y^b z^c$ is continuous in $\mathbb{R}^3$ where $a,b,c$ are whole numbers.
My try:
We need to find a relation between $\epsilon$ and $\delta$.
We are given:
$\sqrt{(x-x_0)^2 + (y-y_0)^2 + (z-z_0)^2} < \delta$
We need to show:
$| x^a y^b z^c - (x_0)^a (y_0)^b (z_0)^c | \leq A [(x-x_0)^2 + (y-y_0)^2 + (z-z_0)^2] < (A\ \delta ^2 = \epsilon) $
What shall be $A$?
I tried to figure out what $A$ will be for a long time and then gave it up. Can anybody here help?