Summary of Limits and Continuity

To study limits and continuity for functions of two variables, we use a $\delta$ disk centered around a given point.
A function of several variables has a limit if for any point in a $\delta$ ball centered at a point $P$ , the value of the function at that point is arbitrarily close to a fixed value (the limit value).
The limit laws established for a function of one variable have natural extensions to functions of more than one variable.
A function of two variables is continuous at a point if the limit exists at that point, the function exists at that point, and the limit and function are equal at that point.

Glossary

boundary point: a point $P_{0}$ of $R$ is a boundary point if every $\delta$ disk centered around $P_{0}$ contains points both inside and outside $R$

connected set: an open set $S$ that cannot be represented as the union of two or more disjoint, nonempty open subsets

interior point: a point $P_{0}$ of $R$ is a boundary point if there is a $\delta$ disk centered around $P_{0}$ contained completely in $R$

$\delta$ ball: all points in $\mathbb{R}^{3}$ lying at a distance of less than $\delta$ from $(x_{0},y_{0},z_{0})$