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Unformatted text preview: Problem 5: Let f;g : R ! R be continuous functions. (a) Prove that if f ( r ) = 0 for all r 2 Q , then f ( x ) = 0 for all x 2 R . (Hint: Proof by contradiction.) (b) Prove that if f ( r ) = g ( r ) for all r 2 Q , then f ( x ) = g ( x ) for all x 2 R . Problem 6: Let f : R ! R be an additive function, i.e., f satises the functional equation: f ( x + y ) = f ( x ) + f ( y ) for any x;y 2 R . Furthermore, suppose that f has at least one point of continuity c 2 R . Prove that, in fact, f is continuous on the entire real line....
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- Fall '07