functionnal equation

**tize** · August 14th 2006, 10:22 AM

Hello !
I have a problem to solve this question :
Find all real continuous functions f verifing : f(x+1)=f(x)+f(1/x)
Have you ever seen this, could you help me please ?

**ThePerfectHacker** · August 14th 2006, 11:49 AM

There is something that bothers me. You say that [tex]f[tex] is countinous on the number line. Yet, it is undefined for any negative integer!

**CaptainBlack** · August 19th 2006, 03:16 AM

Originally Posted by tize

Hello !
I have a problem to solve this question :
Find all real continuous functions f verifing : f(x+1)=f(x)+f(1/x)
Have you ever seen this, could you help me please ?

Can you tell us some of background to this equation,
its context etc?

RonL

**tize** · August 19th 2006, 11:12 AM

I would like but I don't know anything about this function except that it is continuous. Somebody ask me about such a function and I don't know where does it come from ...

**ThePerfectHacker** · August 19th 2006, 04:46 PM

Originally Posted by tize

I would like but I don't know anything about this function except that it is continuous. Somebody ask me about such a function and I don't know where does it come from ...

What makes it unusual that is a composition of,
$f(1/x)$ which usually are not solved for in functional equations.

**Jameson** · August 19th 2006, 08:25 PM

Like PH said, this has a cyclical property to it and try replacing all x's with 1/x. The right side of the equation yields the same thing, showing that $f(\frac{1}{x}+1)=f(x+1)$ .

**tize** · August 20th 2006, 11:57 PM

Yes we can already proove that :
$f(0)=f(\Phi^{-1})=f(-\Phi)=0$ where $\Phi$ is the Gold number : $\frac{1+\sqrt 5}{2}$
and
$f$ has a limit when $x\rightarrow\infty$ , $f\rightarrow f(1)$

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