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

[holezch]

Homework Statement

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

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

 

[VeeEight]

Try finding a sequence (x_n) of terms that converge to irrational x but f(x_n) does not converge to f(x). Take rational values in the sequence.

 

[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

 

[Office_Shredder]

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)?

 

[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?

 

[holezch]

hi guys, thanks!

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

 

[Office_Shredder]

hi guys, thanks!

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

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

 

[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

 

[Office_Shredder]

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))