hw26solutions - 2, and thus, Z b a f dx L ( P,f ) m ( y-x )...

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Math 521 - Advanced Calculus I Instructor: J. Metcalfe Due: April 19, 2010 Assignment 26 1. Suppose f 0, f is continuous on [ a,b ], and that R b a f ( x ) dx = 0. Prove that f ( x ) = 0 for all x [ a,b ]. Suppose not. That is, suppose that f ( x ) = c > 0 for some c [ a,b ]. By continuity, there is a neighborhood V δ ( c ) [ a,b ] so that if y V δ ( c ) then | f ( y ) - c | < c/ 2. (That is, we are applying the definition of continuity with ε = c/ 2.) Thus, f ( y ) > c/ 2 for y V δ ( c ). Let P be a partition containing two points of V δ ( c ), say x,y V δ ( c ) with y > x , as well as a,b . Then, since f 0, L ( P,f ) m ( y - x ) where m is the minimum of f over [ x,y ]. By the above, however, m > c/
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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 ) &gt; . 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 &lt; 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.

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