Continuity of a Function

    Continuity of a function at a point is defined and extended to intervals. Several examples are given to illustrate what can go wrong with continuity; for example, "poles", "jumps", "holes", or some type of oscillating behavior. Then properties of continuous functions and one-sided continuity are discussed. This topic also illustrates how to define a function so that continuity on an interval or at a point can be assured.  

    It should be noted that continuity can not be determined by a graph, even though graphing can help for many functions. Continuity can be checked at each point using the following definition.

Definition (Continuity) A function is continuous at a point means

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

    A function that is not continuous at is said to be discontinuous at or said another way, has a discontinuity at A function is continuous on an open interval precisely when it is continuous at every point in the interval.

Example (Discontinuity) Find three examples of how a discontinuity might arise.

    Solution. First, the function is discontinuous at because is not defined. So one type of discontinuity is a "hole" in the function.
    Secondly, the function
    


is discontinuous at because and thus does not exist and so has a discontinuity at This type of discontinuity is called a "jump".
    Thirdly, the function
    
    

is discontinuous at because is not defined and this type of discontinuity is called a "pole" because as .

• print this page
• pages linking here
• related pages

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