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?

If you allow discontinuous functions, then the general solution is that you pick some subset $A$ of $\Bbb{R}$ and define a function $f_A : \Bbb{R} \to \Bbb{R}$ by:
$$
f_A(x) = \left\{\begin{array}{l@{\quad}l} 1 & \mbox{if $x \in A$} \\
-1 & \mbox{if $x \not\in A$}\end{array}\right.
$$
Any solution is $f_A$ for some unique $A$.
The two continuous solutions correspond to $A = \Bbb{R}$ and $A = \emptyset$.
