Math 301 Problem Set 9 Solutions

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

This preview shows pages 1–3. Sign up to view the full content.

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
This is the end of the preview. Sign up to access the rest of the document.

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 ) ....
View Full Document

{[ snackBarMessage ]}

### Page1 / 6

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

This preview shows document pages 1 - 3. Sign up to view the full document.

View Full Document
Ask a homework question - tutors are online