Math 301 Problem Set 9 Solutions

Math 301 Problem Set 9 Solutions - Math 301/ENAS 513...

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Unformatted text preview: Math 301/ENAS 513: Homework 9 Solution Triet M. Le 1. Page 89: 8, 10, 11, 13. 15. 2. Page 91: 3,6. 3. Page 94: 2, 3, 6. Page 89: 8: B ⊂ S being not closed implies B does not contain all its limit points. Let a ∈ S be a limit point of B such that a / ∈ B . Now, consider f : B → R , with f ( x ) = 1 d ( x,a ) > 0, for all x ∈ B . To show f is continuous: For any x, y ∈ B , | f ( x )- f ( y ) | = 1 d ( x, a )- 1 d ( y, a ) = d ( y, a )- d ( x, a ) d ( x, a ) d ( y, a ) Suppose d ( y, a ) ≥ d ( x, a ), then | d ( y, a )- d ( x, a ) | = d ( y, a )- d ( x, a ) ≤ d ( y, x ) + d ( x, a )- d ( x, a ) = d ( x, y ) On the other hand if d ( x, a ) ≥ d ( y, a ), then | d ( y, a )- d ( x, a ) | = d ( x, a )- d ( y, a ) ≤ d ( x, y ) + d ( y, a )- d ( y, a ) = d ( x, y ) Thus, in either cases, | f ( x )- f ( y ) | ≤ d ( x, y ) d ( x, a ) d ( y, a ) , ∀ x, y ∈ B. Let β x = d ( x, a ) / 2 > 0. Pick any > 0, we have | f ( x )- f ( y ) | ≤ d ( x, y ) d ( x, a ) d ( y, a ) ≤ d ( x, y ) β 2 x < , whenever d ( x, y ) < δ x = min { β x , β 2 x } . Hence f is continuous at any x ∈ B . To show f is not uniformly continuous: Suppose f is uniformly continuous. I.e. for each > 0, there exsists δ > 0 such that | f ( x )- f ( y ) | < whenever d ( x, y ) < δ . Since a is a 1 limit point of B , then for each β ∈ (0 , 1), we can find x, y ∈ B such that d ( y, a ) < βδ and d ( x, a ) < d ( y, a ) / 2. Note f ( x ) > f ( y ) and d ( x, y ) < δ . On the other hand, f ( x )- f ( y ) = 1 d ( x, a )- 1 d ( y, a ) ≥ 2 d ( y, a )- 1 d ( y, a ) = 1 d ( y, a ) ≥ 1 βδ → ∞ as β → ∞ . (1) This is a contradition. Page 89: 10: Let a ∈ R be any irrational number such that a > 0, and define B = [0 , a ] ∩ Q . It is clear that B contains all its limit points in Q . Hence it is closed in Q . Define f : B → R with f ( x ) = 1 / ( x- a ). By the above problem, f is continuous on B . However, | f ( x ) | → ∞ as x → a and x ∈ B , which shows that f is unbounded. Page 89: 11: Let ( S, d S ) and ( T, d T ) be metric spaces with d S being discrete. Let f : S → T be any function. To show f is uniformly continuous: For any > 0. Let δ = 1 / 2. Then N δ ( x ) = { x } for all x ∈ S . Hence | f ( x )- f ( y ) | = 0 < , for all y ∈ N δ ( x ) ....
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Math 301 Problem Set 9 Solutions - Math 301/ENAS 513...

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