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09.28 - [Hand out time sheets Prof Tibor Beke will...

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[Hand out time sheets] Prof. Tibor Beke will substitute for me on Thursday and Friday of this week. Section 1.5: Continuity Key concept: Continuity. A continuous function is one that satisfies the direct substitution property lim x a f ( x ) = f ( a ). Main points of the section? ..?. . Main point #1: Most of the functions normally encountered in pre-calculus math are continuous. Main point #2: Most ways of combining two continuous functions give functions that are continuous too. Main point #3: If a function f is continuous, it satisfies the intermediate value theorem. Definition: We say the function f ( x ) is continuous at a iff (1) lim x a f ( x ) = f ( a ), i.e., in symbols, f ( x ) f ( a ) as x a . Otherwise, we say f ( x ) is discontinuous at a .

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If a function is continuous at every point in its domain, we say that it is everywhere continuous , or (just) continuous . Example 1: One of our limit laws tells us that (for all values a ) lim x a x 2 = a 2 , so the function f ( x ) = x 2 is continuous at a for every a ; that is, it is continuous at every point in its domain (in this case, the domain is R , the set of all real numbers). Example 2: The function sqrt( x ), whose domain is [0, ), is not continuous at 0, because it is not defined throughout a punctured interval { x : 0 < | x –0| < r }, no matter how small r is. Example 3(a): The Heaviside function H ( x ) is not continuous at 0 (though it is continuous at every other value of x ) because lim x 0 H ( x ) does not exist (more specfically, the one-sided limits lim x 0+ H ( x ) and lim x 0– H ( x ) exist but
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09.28 - [Hand out time sheets Prof Tibor Beke will...

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