Determining Continuity

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

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

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:
    

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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Determining Continuity
Published by Library of Math -- Online math organized by subject into topics.
Written by Smith, David A.
http://www.libraryofmath.com/determining-continuity.html