hmw9s - 18.100B Fall 2002 Homework 9 Due by Noon Tuesday...

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± 18.100B, Fall 2002, Homework 9 Due by Noon, Tuesday November 26 Rudin: (1) Chapter 6, Problem 12 Proof. Suppose that f ∈R ( α ) , let C> 0 be such that | f ( x ) |≤ C for all x [ a, b ] . Given ±> 0 there exists a partition P of [ a, b ]suchthat n (1) U ( f, α, P ) L ( f, α, P )= ( α ( x i ) α ( x i 1 ))( M i m i ) 2 / 2 C i =1 where M i and m i are the supremum and infimum of f over [ x i 1 ,x i ] . Con- sider the function given in the hint: t x i 1 (2) g ( t )= x i t f ( x i 1 )+ f ( x i ) ,t [ x i 1 ,x i ] . x i x i 1 x i x i 1 Note that the value at t = x i is independent of choice even if there are two intervals of which it is an end point. On [ x i 1 ,x i ] ,g is continuous since it is linear there and it is continuous at each x i , hence is continuous everywhere. On [ x i 1 ,x i ] ,g takes values in [ m i ,M i ] since its maximum and minimum occur at the ends (it is linear) and these are
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hmw9s - 18.100B Fall 2002 Homework 9 Due by Noon Tuesday...

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