TITLE: Interesting functional equation: $f(x)=\frac{x}{x+f\left(\frac{x}{x+f(x)}\right)}$
QUESTION [8 upvotes]: Solve for the function f(x):
$$f(x)=\frac{x}{x+f\left(\frac{x}{x+f(x)}\right)}$$
I'm not able to solve this. 
[For instance, I tried solving for $f(\frac{x}{x+f(x)})$, but this doesn't lead me anywhere as the value obtained, when substituted into the original equation, just yields $f(x)=f(x)$]
Question: What is f(x) = ?
I suppose I need to mention that f(x) should be continuous, and that it's  a subset of $R$.
(Related : Functional equation $f(x)=\frac{1}{1+f(\frac{1}{1+f(x)})}$ , whose confirmed solutions are $-\phi$ and $\frac{1}{\phi}$)

REPLY [6 votes]: If such an $f$ exists, plugging in $x=0$ we find that
$$f(0)=\frac{0}{0+f\left(\tfrac{0}{0+f(0)}\right)}=0,$$
and to avoid having to divide by $0$ this requires that
$$0+f(0)\neq0\qquad\text{ and }\qquad0+f\left(\frac{0}{0+f(0)}\right)\neq0.$$
But then $f(0)=0$ so $0+f(0)=0$, a contradiction. Hence no such $f$ exists.