Characterization of continuous functions with the property $f(x) = f\left(\frac{x}{1-x}\right)$ [closed]
Determine all the real valued functions $f$ defined on set of reals satisfying:
- $f$ is continuous at $0$, and
- $f(x) = f\left(\frac{x}{1-x}\right)$ for all $x$ other than $1$.
Tried but don't know how to proceed?
Prove by induction that $f(x)=f(\frac x {1-nx})$. Let $n \to \infty$ to conclude that $f(x)=f(0)$ for all $x \neq 1,\frac 1 2,\frac 1 3,...$. You can also check that $f(\frac 1n )=f(\frac 1{n-1})$ for $n >1$. What do you conclude from this now?