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Unformatted text preview: I = (0 , 1), and discontinuous at each irrational point in I . Hint. Show that if f ( x ) is continuous at each rational point in I = (0 , 1), then there exists an irrational number x ∈ I and monotone sequences a j % x and b j & x , such that osc Δ j f ( x ) := sup Δ j finf Δ j f → as j → ∞ , where Δ j := [ a j , b j ] . 5. Let μ be a function deﬁned on all subsets of a ﬁxed set X , such that 0 ≤ μ ( A ) < ∞ for each A ⊆ X , and μ ( A ∪ B ) = μ ( A ) + μ ( B ) if A ∩ B = ∅ . The symmetric diﬀerence of the sets A and B (see p. 3) A Δ B = ( A \ B ) ∪ ( B \ A ) = ( A ∪ B ) \ ( A ∩ B ) . Show that ρ ( A, B ) := μ ( A Δ B ) satisﬁes the triangle inequality ρ ( A, C ) ≤ ρ ( A, B ) + ρ ( B, C ) . Hint. First show that ( A \ C ) ⊆ ( A \ B ) ∪ ( B \ C )....
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This note was uploaded on 02/15/2012 for the course MATH 8601 taught by Professor Staff during the Fall '08 term at Minnesota.
 Fall '08
 Staff
 Math

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