Continuity Function

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

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

(2) 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 .

(3) Proposition (Continuous Functions) If is a polynomial function, rational function, trigonometric function, or inverse trigonometric function, then is continuous where it is defined.

(4) Proposition (Properties of Continuous Functions) If and are functions that are continuous at then and are continuous at provided that is in the domain of the function.

(5) Example (Continuous Functions) Give some examples of continuous functions.

    Solution. For example, the functions (polynomial), (rational), (trigonometric), and (inverse trigonometric) are continuous on their domains. Also the functions and are continuous functions on their domains.

(6) Proposition (Composition Limit Rule) If and is a continuous function at then

(7) Example (Composition Limit Rule) Use the Composition Limit Rule to evaluate the following limits.

        (a)

    Solution. By the Composition Limit Rule, we have
    
        (b)  

    Solution. By the Composition Limit Rule, we have

(8) Definition (One-Sided Continuity) The function is continuous from the right at if and only if and it is continuous from the left at if and only if

(9) Example (One-Sided Continuity) Give an example of a function that is continuous from the right at

    Solution. The function is continuous from the right at because

(10) Definition (Continuity on Interval) A function is continuous on an open interval if it is continuous at every number in the interval, and a function is continuous on if it is continuous on and continuous from the right at Also is continuous on if it is continuous on and continuous from the left at Similarly, for    and

(11) Example (Continuity on an Interval) Find constants and so that



is continuous on

    Solution. Since is defined on and is continuous for all for any and that we choose, it is left to find an and such that



Thus we have the system and Solving this system we have, and

(12) Example (Determining Continuity) Determine the value for which should be assigned, if any, to have continuous at

    Solution. Since and we have Therefore, if we define the function will be continuous at Here's the graph:
    

(13) Example (Determining Continuity) Find constants and such that is continuous at where
    


    Solution. To have continuity at we must have and thus and Therefore, and So and

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