iflugkiwmu wrote: Determine all continuous functions $f : \mathbb{R} \setminus \{1\} \to \mathbb{R}$ that satisfy \[ f(x) = (x+1) f(x^2), \]for all $x \in \mathbb{R} \setminus \{-1, 1\}$. Let $g(x)$ from $\mathbb R\setminus\{1\}\to\mathbb R$ defined as $g(x)=(x-1)f(x)$ Equation becomes $g(x)=g(x^2)$ $\forall x\ne \pm 1$ So $g(-x)=g(x^2)=g(x)$ $\forall x\ne \pm 1$ $\forall x\in[0,1)$ : $g(x)=g(x^{2^n})$ and setting $n\to+\infty$ and using continuity at zero: $g(x)=g(0)$ $\forall x\in[0,1)$ So $g(x)=g(0)$ $\forall x\in(-1,1)$ and continuity at $-1$ implies $g(x)=g(0)$ $\forall x\in[-1,1)$ So $\lim_{x\to -1^-}g(x)=g(-1)=g(0)$ Then, $\lim_{x\to 1^+}g(x)=g(-1)=g(0)$ And, since $\forall x>1$, $g(x)=g(x^{2^{-n}})$ , we get $g(x)=g(0)$ $\forall x\in(1,+\infty)$ and so $g(x)=c$ constant $\forall x\ne 1$ And so $\boxed{f(x)=\frac c{x-1}\quad\forall x\ne 1}$, which indeed fits