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

1. Apr 25, 2010

holezch

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. Apr 25, 2010

VeeEight

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.

3. Apr 25, 2010

holezch

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

4. Apr 25, 2010

Office_Shredder

Staff Emeritus
You know that a+h in decimal form is $$.a_1 a_2 a_3.... a_n = \frac{a_1 a_2 a_3... a_n}{10^{-n}}$$. Now that you know what a+h is as a rational number, what can you say about f(a+h)?

5. Apr 25, 2010

VeeEight

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?

6. Apr 25, 2010

holezch

hi guys, thanks!

But officeshredder, how do I know that your fraction there is in lowest terms?

7. Apr 25, 2010

Office_Shredder

Staff Emeritus
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

8. Apr 25, 2010

holezch

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

9. Apr 25, 2010

Office_Shredder

Staff Emeritus
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))