Analysis problem

Find a function on the reals which is everywhere continuous but nowhere differentiable.

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

?

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->r+) f(x) = 0 = lim(x->r-) f(x) < f(x) = 1.

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.

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 ?

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

Originally posted by Nuathala
This example was given by Weierstrass in 1861

That's cheating! 😠

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.

surely there are more beautiful solutions?

Originally posted by TheMaster37
surely there are more beautiful solutions?

What was the one Acolyte had in mind?

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!

Product/Quotient rule?

(sin(x) / x)' = (sin(x))' / x + sin(x) * (1/x)' = cos(x) / x - sin(x) / x^2

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 😛 😳

In the stochastic calculus there are functions called wiener processes. Those are almost surely nowhere differentiable. "almost surely" 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 😀

i need to take calc.