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Unformatted text preview: , (1) f ( xy ) = f ( x ) f ( y ) . (2) Claim: f ( x ) = 0 for all x or f ( x ) = x for all x . Prove this claim using the following steps: (a) Prove that f (0) = 0 and that f (1) = 0 or 1. (b) Prove that f ( n ) = nf (1) for every integer n and then that f ( n/m ) = ( n/m ) f (1) for all integers n,m such that m 6 = 0. Conclude that either f ( q ) = 0 for all rational numbers q or f ( q ) = q for all rational numbers q . 1 (c) Prove that f is nondecreasing, that is to say that f ( x ) f ( y ) whenever x y for any real numbers x and y . (d) Prove that if f (1) = 0 then f ( x ) = 0 for all real numbers x . Prove that if f (1) = 1 then f ( x ) = x for all real numbers x . 2...
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This note was uploaded on 02/15/2012 for the course MATH 18.100B taught by Professor Prof.katrinwehrheim during the Fall '10 term at MIT.
- Fall '10