Properties of a Continuous Function

    Even though it is often difficult to determine whether a given function is continuous at a specified number, there are many common functions that are continuous everywhere. Moreover, continuous functions can be combined in various ways without making a discontinuity.

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

    Since the definition of continuity is based on the limit, proofs of the following theorem can be given in terms of the definition of the limit.

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.

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.

    The next theorem (continuous composition property) states that a continuous function of a continuous function is continuous.

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

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

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Properties Of A Continuous Function
Published by Library of Math -- Online math organized by subject into topics.
Written by Smith, David A.
http://www.libraryofmath.com/properties-of-a-continuous-function.html