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By Anthony W. Knapp

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Extra resources for Basic Real Analysis

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10we know that f is uniformly continuous on [a,b ] . Given t > 0, we can therefore choose some number 6 > 0 corresponding to f and t on [a, b ] . Let P = be a partition on [ a ,b] of mesh p ( P ) < 8. On any I. ll. Since Iqi 5 Ixi-xi-lI = Axi 5 p ( P ) < S, we obtain Mi - mi = f (qi)- f (ti) < t. Therefore and thc thcorcm follows from Lcmma 1 . 2 5 ~ . 26 by allowing finitely illany poiiits of discontinuity, but we need to do some additional work beforehand. 27. If f is bounded on [a, b] and a 5 c 5 b, then 1 dx = 1'.

Hence the sum of the corresponding terms of U ( P , f ) is + 5 (kThe sum of the terms of the second kind is 5 U ( P *, f ) . Thus u ( p ,f ) 5 t + U ( P * ,f ) 5 Lb f'dx + 2t. Similarly we can produce 62 such that p ( P ) 5 62 implies I. Theory o f Calculus in One Real Variable 40 If S = min{S1,S2}and p ( P ) 5 8, then lab andhence S ( P , f ) f d x 5 2c. For the converse let t > 0 be given, and choose some S as in the statement of the I -b I theorem. Next choose a partition P = {xi}:=, with U ( P , f ) - Jh f' dx < r and f dx - L I P , f ) < r : possibly by passing to a refinement of P .

The first step is to apply the Mean Value Theorem to f, - f m , estimate f; - f k , and use the convergence of { fn ( x o ) }to obtain the existence of the limit function f . 17. The second step is to apply the Mean Value Theorem again to f , - f m , this time to see that converges uniformly in t (for t # x ) as n tends to infinity, the limit being q ( t ) = (I) . In fact, the Mean Value Theorem produces some strictly between t-x t and x such that 'Pn( t ) - 'Pm ( t ) = [ f n ( t ) - fm(t)l - [ f n ( ~-) fm(x)I = f; t-x 0 )- f:, 0 ) 9 and the right side tends to 0 uniformly as n and m tend to infinity.

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