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: 0 when irrational, 1/q in lowest terms with rational not differentiable.

  1. Apr 25, 2010 #1
    1. The problem statement, all variables and given/known data
    Hi, I have this function:

    f(x ) = 0 (x is irrational) or f(x) = 1/q for rational p/q in lowest terms.

    show that this function is not differentiable anywhere

    3. The attempt at a solution

    This is the answer from the solutions book:
    consider [(f(a+h) - f(a ) ) / h ]. function f is discontinuous at rational numbers, so we only have to look at irrational a.

    if h is rational, then f(a+h) - f(a ) = 0
    if h is irrational, then let a = m.a1a2a3...an and h = -0.00...an+1
    Then, a + h = m1.a1a2a3..an0000 and f(a+h) > 10^(-n)



    My question is.. how do we know anything about f(a+h)? a+h is > 10^(-n) but what is f(a+h)?

    thanks
     
  2. jcsd
  3. Apr 25, 2010 #2
    Try finding a sequence (xn) of terms that converge to irrational x but f(xn) does not converge to f(x). Take rational values in the sequence.
     
  4. Apr 25, 2010 #3
    thanks for the tip.. I will think about it.. (I think it will be some kind of dirchlet type function?) But is that going to help answer the f(a+h) > 10^(-n) confusion?
    thanks
     
  5. Apr 25, 2010 #4

    Office_Shredder

    User Avatar
    Staff Emeritus
    Science Advisor
    Gold Member

    You know that a+h in decimal form is [tex] .a_1 a_2 a_3.... a_n = \frac{a_1 a_2 a_3... a_n}{10^{-n}}[/tex]. Now that you know what a+h is as a rational number, what can you say about f(a+h)?
     
  6. Apr 25, 2010 #5
    No, I just suggested it because I think it is the much easier method.
    To compute f(a+h), write down a+h's decimal expansion. What kind of number is it? Where does f send it to?
     
  7. Apr 25, 2010 #6
    hi guys, thanks!

    But officeshredder, how do I know that your fraction there is in lowest terms?
     
  8. Apr 25, 2010 #7

    Office_Shredder

    User Avatar
    Staff Emeritus
    Science Advisor
    Gold Member

    You don't. But if you can write it on lower terms the denominator is smaller, which makes f(x) bigger

    Also, I'm not convinced such a sequence exists VeeEight. The function seems to be continuous at all irrational points
     
  9. Apr 25, 2010 #8
    I'm a bit confused.. how do you write that fraction into p/q lowest terms where p and q are integers? thanks
    and the function is continuous for all irrational points
     
  10. Apr 25, 2010 #9

    Office_Shredder

    User Avatar
    Staff Emeritus
    Science Advisor
    Gold Member

    We have it in the form a/(10n). This might not be in lowest terms, which means the numerator and denominator share a common factor. How do you write it in lowest terms? You cancel the common factor, making it say p/q. This makes both the numerator and the denominator smaller, so if q<10n, what can you say about 1/q (which is f(a/10n))
     
Share this great discussion with others via Reddit, Google+, Twitter, or Facebook