a b there exists a partition P x x 2 x n of a b such that U f P L f P \u03b4 2 We

# A b there exists a partition p x x 2 x n of a b such

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a, b ], there exists a partition P = { x 0 , x 2 . . . , x n } of [ a, b ] such that U ( f, P ) - L ( f, P ) < δ 2 . We claim that for this partition we also have U ( g f, P ) - L ( g f, P ) = n i =1 [ M i ( g f ) - m i ( g f )] Δ x i < . To show this, we separate the set of indices of the partition P into two disjoint sets. A = { i : M i ( f ) - m i ( f ) < δ } and B = { i : M i ( f ) - m i ( f ) δ } . Then if i A and x, y [ x i - 1 , x i ], we have | f ( x ) - f ( y ) | ≤ M i ( f ) - m i ( f ) < δ, so that | g f ( x ) - g f ( y ) | < . But then M i ( g f ) - m i ( g f ) . It follows that i A [ M i ( g f ) - m i ( g f )] Δ x i i A Δ x i ( b - a ) . 4
On the other hand, if i B , then [ M i ( f ) - m i ( f )] 1, so that i B Δ x i 1 δ i B [ M i ( f ) - m i ( f )] Δ x i 1 δ [ U ( f, P ) - L ( f, P )] < δ < . Thus since M i ( g f ) - m i ( g f ) 2 K for all i , we have i B [ M i ( g f ) - m i ( g f )] Δ x i 2 K i B Δ x i < 2 K . Now when we combine all the indices we obtain U ( g f, P ) - L ( g f, P ) = i A [ M i ( g f ) - m i ( g f )] Δ x i + i B [ M i ( g f ) - m i ( g f )] Δ x i ( b - a ) + 2 K = ( b - a + 2 K ) < . 5

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• Winter '11
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• Math, Continuous function, Metric space, dt, dx