1. Not finding help here? Sign up for a free 30min 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!

Analysis problem

  1. Nov 11, 2008 #1
    I need to show if the following is true or false.

    If the function f: (0,1)--> R is continuous in every irrational number x then f is continuous at every number.

    Thank you
    1. The problem statement, all variables and given/known data



    2. Relevant equations



    3. The attempt at a solution
     
  2. jcsd
  3. Nov 11, 2008 #2
    The answer is false: Define f by f(x) = 0 if x is irrational; and f(x) = 1/q where x = p/q, p coprime to q. This function is continuous at every irrational and discontinuous at all the rationals.
     
  4. Nov 11, 2008 #3
    There are functions where this is true. f(x) = x is continuous at every irrational number on [0,1], and is continuous at every number. However, this is not generally the case:
    I believe Thomae's function serves as a counterexample to the statement which you need to disprove which is what e(ho0on3 mentioned.

    There is an interesting paper on Thomae's function by Dr. Beanland, discussing how to modify the function so that is differentiable
    <www.people.vcu.edu/~kbeanland/Papers/ThomaesFunction.pdf>
    Very interesting, he mentions some way to quantify irrationalness which is the part of the paper that goes over my head, but is interesting nonetheless
     
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook

Have something to add?



Similar Discussions: Analysis problem
  1. Analysis problem (Replies: 3)

  2. Analysis problem (Replies: 3)

  3. Analysis problem (Replies: 2)

  4. Analysis problem (Replies: 2)

  5. A analysis problem (Replies: 1)

Loading...