Summary of Limits and Continuity

Essential Concepts

  • To study limits and continuity for functions of two variables, we use a δ disk centered around a given point.
  • A function of several variables has a limit if for any point in a δ 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 P0 of R is a boundary point if every δ disk centered around P0 contains points both inside and outside R
closed set
a set S that contains all its boundary points
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 P0 of R is a boundary point if there is a δ disk centered around P0 contained completely in R
open set
a set S that contains none of its boundary points
region
an open, connected, nonempty subset of R2
δ ball
all points in R3 lying at a distance of less than δ from (x0,y0,z0)
δ disk
an open disk of radius δ centered at point (a,b)