Summary of Limits and Continuity

To study limits and continuity for functions of two variables, we use a [latex]\delta[/latex] disk centered around a given point.
A function of several variables has a limit if for any point in a [latex]\delta[/latex] ball centered at a point [latex]P[/latex], 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 [latex]P_{0}[/latex] of [latex]R[/latex] is a boundary point if every [latex]\delta[/latex] disk centered around [latex]P_{0}[/latex] contains points both inside and outside [latex]R[/latex]

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

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

region: an open, connected, nonempty subset of [latex]\mathbb{R}^{2}[/latex]

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

[latex]\delta[/latex] disk: an open disk of radius [latex]\delta[/latex] centered at point [latex](a,b)[/latex]