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Unformatted text preview: Homework 12 for MATH 104 Due: Tuesday Dec 2 Problem 1 Pugh, p.196, #51 Solution. (a) True, as we know from the corollary in the textbook (since the absolute value function is continu- ous, so the composite is Riemann integrable). (b) False, for example consider a function f : [0 , 1] R where f ( x ) = 1 if x Q and f ( x ) =- 1 if x < Q . Clearly, | f | is Riemann integrable, since | f | is a constant function (always 1). However, f is not Riemann integrable, since it is discontinuous at every point (analogous to the case of the characteristic function for the rationals). (c) Let f be a Riemann integrable function and | f ( x ) | c > 0 for all x . f is Riemann integrable, so f is bounded. Since | f ( x ) | > 0 for all x , clearly the function 1 / f is bounded. Let D be the set of discontinuity points of f . Since | f ( x ) | > 0, the set of discontinuity points of 1 / f is also D . Since f is Riemann integrable, D is a zero set....
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- Fall '08