Describe all functions which are defined as $f:\mathbb R \to \mathbb R $ and satisfying $f(x)=\frac1{f(x)}$ Describe all functions which are defined as $f:\mathbb R \to \mathbb R $ and satisfying $$f(x)=\dfrac1{f(x)}$$
My Attempt:
If $f$ is constant function then.
$$(f(x))^2=1$$
Then $f(x)=1$ or $f(x)=-1$
I cannot think non-constant function satisfying this
What about $f$ being continuous?

$f$ satisfies this equation iff there exists a set $A$ such that $f =2\chi_A-1$. If $f$ is continuous then it is identically $1$ or identically $-1$.
[For the proof define $A$ as $\{x: f(x)=1\}$; $ \chi_A$ is defined by $ \chi_A(x)=1$ if $ x \in A$ and $0$ if $x \notin A$].
