# Thread: functionnal equation

1. ## functionnal equation

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 ?

2. 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!

3. 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

4. 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 ...

5. 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,
$\displaystyle f(1/x)$ which usually are not solved for in functional equations.

6. 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 $\displaystyle f(\frac{1}{x}+1)=f(x+1)$.

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