Intro to Analysis in-class_Part_30

Intro to Analysis in-class_Part_30 - 4.2 Properties of...

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4.2 Properties of continuity 65 4.2 Properties of continuity Theorem 4.2.1. If f,g are continuous, then so are f + g , f · g , and (if g 6 = 0 ) f/g . Proof. sequences. NOTE: Define ( f + g )( x ) := f ( x ) + g ( x ), etc. USE this with the thm that lim commutes with continuous functions: Example 4.2.1. NOTE: all limits are finite, and the denom 6 = 0! lim t 1 3 t 2 - t t 2 + 1 = lim t 1 (3 t 2 - t ) lim t 1 ( t 2 + 1) = 3lim t 1 t 2 - lim t 1 t lim t 1 t 2 + lim t 1 1 = 3 - lim t 1 t 1 + 1 = 2 2 = 1 Theorem 4.2.2. Let x = g ( t ) ,c = g ( b ) . If g ( t ) is continuous at b and f ( x ) is continuous at c , then f g ( t ) = f ( g ( t )) is continuous at b . Proof. Given ε > 0 , δ > 0 such that f ( x ) ε f ( c ) for x δ c continuity of f g ( t ) δ g ( b ) for t α b continuity of g. Then
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This note was uploaded on 11/26/2011 for the course MAT 4944 taught by Professor Wong during the Fall '11 term at FSU.

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Intro to Analysis in-class_Part_30 - 4.2 Properties of...

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