Does this functional equation have a solution (also is there free functional equation solving software)?

Question

I tried typing this into Wolfram and it couldn't solve it, despite it looking simple:

$f(x) = (1/2)f(x)^2.$

Does this have a non-constant solution? (initial values not yet determined)

Answer 1

Why resort to software, though? The functional notation is confusing you. Instead of $f(x)$ just use e.g $z$. We have the equation $$z=\frac{z^2}{2}$$ The only solutions for this are $z=0$ and $z=2$. This means the only possible solutions for $f$ are $$f:x\mapsto 0 \\ f:x\mapsto 2$$ Good luck finding a non-constant solution, because they don't exist.

EDIT: I spoke to soon. If $f$ attains only the values $0$ or $2$ it will work. E.g,

$$f(x)=\begin{cases} 0 & x< 0\\ 2 & x\geq 0 \end{cases}$$ Works.