Intro to Analysis in-class_Part_40

# Intro to Analysis in-class_Part_40 - 6.1 Integrals of...

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6.1 Integrals of continuous functions 85 Theorem 6.1.12. Let f ∈ R [ a,b ] ,m f M . If ϕ is continuous on [ m,M ] and h ( x ) := ϕ ( f ( x )) for x [ a,b ] , then h ∈ R [ a,b ] . Proof. Fix ε > 0. Since ϕ is uniformly continuous on [ m,M ], ﬁnd 0 < δ < ε such that | s - t | < δ = ⇒ | ϕ ( s ) - ϕ ( t ) | < ε, s,t [ m,m ] . Since f ∈ R , ﬁnd P = { x 0 ,...,x n } such that Osc ( f,P ) < δ 2 . M i ,m i are extrema of f , M 0 i ,m 0 i are for h . Subdivide the set of indices { 1 ,...,n } into two classes: i A ⇐⇒ M i - m i < δ, i B ⇐⇒ M i - m i δ. For i A , have M 0 i - m 0 i ε by choice of δ . For i B , have M 0 i - m 0 i 2sup m t M | ϕ ( t ) | . By prev bound of δ 2 , δ X i B ( x i - x i - 1 ) X i B ( M i - m i )( x i - x i - 1 ) < δ 2 X i B ( x i - x i - 1 ) < δ. Then Osc ( h,P ) = X i A ( M 0 i - m 0 i )( x i - x i - 1 ) + X i B ( M 0 i - m 0 i )( x i - x i - 1 ) ε ( b - a ) + 2 δ sup | ϕ ( t ) | < ε ( b - a + 2sup

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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_40 - 6.1 Integrals of...

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