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Unformatted text preview: 2, and thus, Z b a f dx L ( P,f ) m ( y-x ) ( c/ 2)( y-x ) > . This proves the contrapositive. 2. If f ( x ) = 0 for all irrational x , f ( x ) = 1 for all rational x , prove that f 6 R on [ a,b ] for any a < b . Fix any partition P . Since there is a rational between any two real numbers, U ( P,f ) = 1. On the other hand, since there is an irrational between any two real numbers, L ( P,f ) = 0. Since the partition was arbitrary, R b a f dx = 1 6 = 0 = R b a f dx. 1...
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This note was uploaded on 07/15/2010 for the course MATH 521 taught by Professor Metcalfe during the Spring '10 term at University of North Carolina Wilmington.
- Spring '10