Essential Concepts
- To study limits and continuity for functions of two variables, we use a disk centered around a given point.
- A function of several variables has a limit if for any point in a ball centered at a point , 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 of is a boundary point if every disk centered around contains points both inside and outside
- closed set
- a set that contains all its boundary points
- connected set
- an open set that cannot be represented as the union of two or more disjoint, nonempty open subsets
- interior point
- a point of is a boundary point if there is a disk centered around contained completely in
- open set
- a set that contains none of its boundary points
- region
- an open, connected, nonempty subset of
- ball
- all points in lying at a distance of less than from
- disk
- an open disk of radius centered at point
Candela Citations
CC licensed content, Shared previously
- Calculus Volume 3. Authored by: Gilbert Strang, Edwin (Jed) Herman. Provided by: OpenStax. Located at: https://openstax.org/books/calculus-volume-3/pages/1-introduction. License: CC BY-NC-SA: Attribution-NonCommercial-ShareAlike. License Terms: Access for free at https://openstax.org/books/calculus-volume-3/pages/1-introduction