This preview shows pages 1–2. Sign up to view the full content.
This preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
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....
View
Full
Document
 Fall '08
 RIEMAN
 Math

Click to edit the document details