lec16

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

This preview shows page 1. Sign up to view the full content.

This is the end of the preview. Sign up to access the rest of the document.

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...
View Full Document

## 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.

Ask a homework question - tutors are online