lec16 - Math 117 – May 26, 2011 16 Continuous functions...

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Unformatted text preview: Math 117 – May 26, 2011 16 Continuous functions Definition 16.1. Let f : D → R and let c ∈ D . We say that f is continuous at c , if for every > there exists δ > 0 such that | f ( x )- f ( c ) | < , whenever | x- c | < δ and x ∈ D . If f is continuous at every point of a subset S ⊂ D , then f is said to be continuous on S . If f is continuous on its domain D , then f is said to be continuous . Notice that in the above definition, unlike the definition of limits of functions, c is not required to be an accumulation point of D . It is clear from the definition that if c is an isolated point of the domain D , then f must be continuous at c . The following statement gives the interpretation of continuity in terms of neighborhoods and se- quences. Theorem 16.2. Let f : D → R and let c ∈ D . Then the following three conditions are uqeivalent. ( a ) f is continuous at c ( b ) If ( x n ) is any sequence in D such that x n → c , then lim f ( x n ) = f ( c ) ( c ) For every neighborhood...
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This note was uploaded on 05/31/2011 for the course MATH 117 taught by Professor Akhmedov,a during the Spring '08 term at UCSB.

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