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!

Check my solution for discontinuous function

  1. Dec 6, 2013 #1
    1. The problem statement, all variables and given/known data
    Let f: R -> R and defined with f(x)={ x, if x [itex]\in[/itex] Q or x^2 if [itex]\in[/itex] R\Q}
    a) Prove that function is discontinuous at x=2;
    b) Find all points for R in which function is continuous;

    3. The attempt at a solution
    As far as i know there are infinitely many irrationals but more importantly i can find infinitely many irrational numbers around any number. If that's the case then:

    a) In a i need to show that function is discontinuous, so i need to find ε>0 such that for all δ>0, and for some x, i have | x - 2| < δ and | f(x) - 2 | > ε

    So if i let ε= 1/2, i can find for any δ, no matter how small, a number x that is irrational. When number is irrational f(x)=x^2. Thus if x>2 and irrational, f(x) is going to be a bit above 4 for values close to 2 and then | f(x) - 2 | approximately 2 which is greater then ε= 1/2, and this works for any δ. Thus it is discontinuous at x=2.

    b) If instead of x=2 i let x=q where, q is [itex]\in[/itex] Q, i can use the same proces as in a to show that it is discontinuous at any q, where q is in Q. Also if i let x be irrational i can do same thing again and show that function is not continuous at any irrational number. So this means that it is not continuous at any point .
     
  2. jcsd
  3. Dec 6, 2013 #2
    what about the continuity at x = 0? and x = 1?
     
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook

Have something to add?
Draft saved Draft deleted



Similar Discussions: Check my solution for discontinuous function
  1. Check my solution. (Replies: 1)

  2. Check my solution (Replies: 2)

Loading...