1. Limited time only! Sign up for a free 30min personal tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Homework Help: Continuity characterization (metric spaces)

  1. May 30, 2009 #1
    1. The problem statement, all variables and given/known data

    Let (X,d) and (Y,d') be metric spaces and f: X-> Y a continuous map.
    Suppose that for each a>0 there exists b>0 such that for all x in X
    we have:

    B(f(x), b) is contained in closure( f(B(x,a))).

    Here B(f(x),b) represents the open ball with centre f(x) and radius b.
    Similarly B(x,a) represents the open ball with centre x and radius a.

    Prove that for all x in X and for every c > a :

    B(f(x), b) is contained in f(B(x,c)).

    3. The attempt at a solution

    No clue here, I took an y element in B(f(x),b) so d(f(x),y) < b.
    Then by assumption B(f(x),b) is contained in closure(f(B(x,a)) so y
    is in closure(f(B(x,a)), and then?
  2. jcsd
  3. May 30, 2009 #2


    User Avatar

    If you don't get an answer, this forum

    http://www.mathhelpforum.com/math-help/ [Broken]

    should help you immensely.

    Questions in their section on analysis are typically very well answered.
    Last edited by a moderator: May 4, 2017
  4. May 31, 2009 #3

    matt grime

    User Avatar
    Science Advisor
    Homework Helper

    This is ripe for a proof by contradiction approach (which may even yield a direct proof).

    If there is a c such that B(f(x),b) is not contained in f(B(x,c)) what does that mean?
  5. May 31, 2009 #4
    Hi Matt, thanks for your reply. Assume there exists a point q in B(f(x),b) such
    that q is not in f(B(x,c)).

    Now since c>a it follows that B(x,a) is contained in B(x,c).
    Hence f(B(x,a)) is contained in f(B(x,c)).

    Since q is not in f(B(x,c)) then q is not in f(B(x,a)).

    But f is continuous so f(closure(B(x,a))) is contained in closure(f(B(x,a)).

    I'm stuck here. What else should I do?
Share this great discussion with others via Reddit, Google+, Twitter, or Facebook