Continuity of Multivariate Functions

Definition (Continuity of Multivariate Functions) The function is continuous at the point if

        (i) is defined,
    
        (ii) exists, and
    
        (iii)

A function is continuous on a set if it is continuous at every point in the set.

Definition (Continuity of Functions of Two Variables) The function is continuous at the point if

        (i) is defined,
    
        (ii) exists, and
    
        (iii)

A function is continuous on a set if it is continuous at every point in the set.

    The function is continuous at if the functional value of is close to whenever , in the domain of is sufficiently close to Geometrically, this mean that is continuous if the surface has no "gaps" or "holes".
    The basic properties of limits can be used to show that if and are both continuous on a set, then so are and where is defined.

Example (Continuity of Functions of Two Variables) On what set is the function



continuous?  

    Solution. We know is continuous for since is a rational function. Since we have that is continuous on .
    
Example (Continuity of Functions of Two Variables) On what set is the function continuous?

    Solution. Let and . Then



Now is continuous everywhere since it is a polynomial and is continuous on its domain . Thus, is continuous on its domain



which consists of all points outside the circle
  

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Cite this as:
Continuity Of Multivariate Functions
Published by Library of Math -- Online math organized by subject into topics.
Written by Smith, David A.
http://www.libraryofmath.com/continuity-of-multivariate-functions.html