P2 - 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

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Math 8601: REAL ANALYSIS. Fall 2010 Homework #2 (due on W, September 29). 50 points are divided between 5 problems, 10 points each. #1. Let f ( x ) be a continuous function on [ - 1 , 1], such that f ( - 1) < 0 < f (1) . Show that f ( c ) = 0 for some c ( - 1 , 1). Hint. Take c := sup { x [ - 1 , 1] : f ( x ) 0 } . 2. Let ( X 1 , ρ 1 ) be a compact metric space, i.e. any sequence { x j } ⊂ X 1 contains a convergent subsequence { x j k } → x 0 X 1 , which means ρ 1 ( x j k , x 0 ) 0 as k → ∞ . Let f be a continuous mapping from ( X 1 , ρ 1 ) to another metric space ( X 2 , ρ 2 ). Show that f is uniformly continuous on X 1 , i.e. ε > 0 , δ > 0 , such that from x, y X 1 ; ρ 1 ( x, y ) < δ it follows ρ 2 ( f ( x ) , f ( y )) < ε. 3. Let f ( x ) be a function defined on the interval I := (0 , 1) in the following way: (i) f ( x ) = 1 /n if x is a rational number of the form x = m/n , where the natural numbers m and n do not have common factors; (ii) f ( x ) = 0 if x is irrational. Show that f ( x ) is continuous at each irrational point in I , and discontinuous at each rational point in I . 4. Show that there are NO functions f ( x ) which are continuous at each rational point in
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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 f-inf 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.

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