Analysis problem

Acolyte
Posers and Puzzles 16 Oct '03 14:35
1. Acolyte
16 Oct '03 14:35
Find a function on the reals which is everywhere continuous but nowhere differentiable.
2. TheMaster37
Kupikupopo!
16 Oct '03 17:47
Originally posted by Acolyte
Find a function on the reals which is everywhere continuous but nowhere differentiable.
Wasn't that something like...

f(x) = 1 for x a rational number
f(x) = 0 for x a non-rational number

?
3. royalchicken
CHAOS GHOST!!!
16 Oct '03 18:371 edit
Originally posted by TheMaster37
Wasn't that something like...

f(x) = 1 for x a rational number
f(x) = 0 for x a non-rational number

?
Your f(x) is discontinuous at every rational x.

If r is a rational, then lim(x-&gt;r+) f(x) = 0 = lim(x-&gt;r-) f(x) &lt; f(x) = 1.
4. royalchicken
CHAOS GHOST!!!
16 Oct '03 20:01
I'd guess it would be most useful to define the function on some interval [a,b) and than have it periodic of period b-a. Then come up with one that works on that interval.

5. 17 Oct '03 16:01
Originally posted by royalchicken
I'd guess it would be most useful to define the function on some interval [a,b) and than have it periodic of period b-a. Then come up with one that works on that interval.

sin(x)/x ?
6. 17 Oct '03 16:04
Originally posted by Acolyte
Find a function on the reals which is everywhere continuous but nowhere differentiable.
take the function f(x)= sum from n=1 to infinity (2^(-n)*cos(8^n*Pi*x))

This example was given by Weierstrass in 1861

Nuathala
7. Acolyte
17 Oct '03 18:07
Originally posted by Nuathala
This example was given by Weierstrass in 1861
That's cheating! ðŸ˜
8. royalchicken
CHAOS GHOST!!!
18 Oct '03 01:171 edit
I made one up. Let f(x) = 1+x for x in [-2,0] and 1-x if x is in [0,2] and continuing elsewhere where f(x+4) = f(x). Then let:

F(x) = SUM (n=1 to infinity) 2^-n f(2^2^n x)

F(x) is everywhere continuous and nowhere differentiable. In general, just find a function with regular points where the derivative fails to exist and add them up in such a way that at least one term is guaranteed to have an undefined derivative. I think many others work.

I've got a proof; thanks for this, twas fun to work out.
9. TheMaster37
Kupikupopo!
19 Oct '03 17:55
surely there are more beautiful solutions?
10. royalchicken
CHAOS GHOST!!!
19 Oct '03 21:50
Originally posted by TheMaster37
surely there are more beautiful solutions?
What was the one Acolyte had in mind?
11. 20 Oct '03 17:48
Originally posted by royalchicken
What was the one Acolyte had in mind?
what was wrong with sin(x)/x ? - or can someone differentiate that? I can't!
12. TheMaster37
Kupikupopo!
20 Oct '03 19:17
Product/Quotient rule?

(sin(x) / x)' = (sin(x))' / x + sin(x) * (1/x)' = cos(x) / x - sin(x) / x^2
13. 21 Oct '03 23:13
Originally posted by TheMaster37
Product/Quotient rule?

(sin(x) / x)' = (sin(x))' / x + sin(x) * (1/x)' = cos(x) / x - sin(x) / x^2
doh ðŸ˜› ðŸ˜³
14. Fiathahel
Artist in Drawing
23 Oct '03 17:04
In the stochastic calculus there are functions called wiener processes. Those are almost surely nowhere differentiable. &quot;almost surely&quot; is because of the randomness of the function. The only problem is that they are too complicated to explain here. But they look like the graph of stockprices. The reason they are nowheree differentiable is because you can construct one taking the sum of tent functions (functions like f(x)=x for x in [0,1] f(x)=-x+1 for x in (1,2] and f(x)=0 elsewhere) Then of course you need to add incountable infinite tent functions.
But it works ðŸ˜€
15. 17 Nov '03 20:25
i need to take calc.