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# Continuity - Continuity We have seen that any polynomial...

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Continuity We have seen that any polynomial function P ( x ) satisfies: for all real numbers a . This property is known as continuity . Definition. Let f ( x ) be a function defined on an interval around a . We say that f ( x ) is continuous at a iff Otherwise, we say that f ( x ) is discontinuous at a . Note that the continuity of f ( x ) at a means two things: (i) exists, (ii) and this limit is f ( a ). So to be discontinuous at a , means (i) does not exist, (ii) or if exists, then this limit is not equal to f ( a ). Basic properties of limits imply the following: Theorem. If f ( x ) and g ( x ) are continuous at a. Then (1) f ( x ) + g ( x ) is continuous at a ; (2)

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is continuous at a , where is an arbitrary number; (3) is continuous at a ; (4) is continuous at a , provided ; (5) If f ( x ) is positive, i.e. , then is continuous at a ; (6) If f ( x ) is continuous at a and g ( x ) is continuous at f ( a ), then their composition is continuous at a . Remark. Many functions are not defined on open intervals. In this case, we can talk about one- sided continuity. Indeed, f ( x ) is said to be
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Continuity - Continuity We have seen that any polynomial...

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