f is a continuous function from R->R and f satisfies f(2x+1)=f(x), then prove that f is constant.

Printable View

- Jun 19th 2010, 08:08 AMChandru1Continuous function
f is a continuous function from R->R and f satisfies f(2x+1)=f(x), then prove that f is constant.

- Jun 19th 2010, 08:14 AMAckbeet
Is f also differentiable?

- Jun 19th 2010, 08:36 AMChandru1
Hi--

No f is not differentiable. Ok, suppose you have an counterexample then please do provide me with one or else, prove the result with differentiability holding true. As far as i know, i dont think f is differntiable. - Jun 19th 2010, 08:40 AMBruno J.
It doesn't need to be differentiable!

Hint : show that is constant on a dense subset of . - Jun 19th 2010, 11:00 AMJG89
If it's of any help, f is differentiable at x = 1 with derivative 0, and f(1) = f(0).

for all non-zero h.

And . - Jun 19th 2010, 07:57 PMChandru1hi
Hi--

How do we prove it for dense subsets. The only way i can think of is prove this true for rationals since they are dense in R.

But i have an other idea. Let g(x)=f(x-1)=> g(2x)=f(2x-1)=f(x-1)=g(x). But by doing this repeatedly we can have g(x)->g(0). Does this work.