continuity.pdf - REAL ANALYSIS UNIFORM CONTINUITY...

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REAL ANALYSIS UNIFORM CONTINUITY Definition f ( x ) is said to be uniformly continuous in a set s , given ε > 0 δ = δ ( ε ) || f ( x 1 ) - f ( x 2 ) | < ε whenever | x 1 - x 2 | < δ and x 1 , x 2 S . Theorem Suppose f ( x ) is continuous in [ a, b ] relative to [ a, b ], then f ( x ) is uniformly continuous in [ a, b ]. Note: False for open intervals. To prove uniform continuity it suffices to prove the following: Given ε > 0 a finite subdivision of [ a, b ] | oscilla- tion f ( x ) < ² x [ x ν - 1 x ν ] or: if M ν , m ν are the upper and lower bounds of f ( x ) in [ x ν - 1 x ν ], we have M ν - m ν < ε ν = 1 , 2 , . . . , n First Proof (using Bisection) Suppose the result is false. Then ε 0 so that for this ε 0 , there is no subdivision of the required type. We call an interval a good interval if bd - bd < ε 0 , and bad otherwise. Subdivide [ a b ] into two equal closed intervals h a, a + b 2 i , h a +2 2 , b i . At least one of these is bad. We define J 1 to be the bad interval if there is only one, or the left hand one if there is a choice. Now subdivide J 1 into two equal intervals as
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