Find all functions $f:\mathbb R \rightarrow \mathbb R$ that have following two properties

Question

Find all functions $f:\mathbb R \rightarrow \mathbb R$ that have following two properties

(i) $f(f(x))=x$ $\;$ $\forall \in \mathbb R$

(ii) $x \geq y$ then $f(x)\geq f(y)$

My Approach:

$f(f(x))=x$ $\implies$ $f(x)$ is one-one and onto.

So let $f(x)=t$

From $(i)$ property

$f(y)=t$

$\implies$

let $t\neq x$

Case $(1)$ $x<t$

$\implies$

$f(x)\leq f(t)$

$\implies$

$t\leq x$

Hence Contradiction.

Case $(2)$ $t<x$

$\implies$

$f(t)\leq f(x)$

$\implies$

$x\leq t$

Hence again contradiction

From Case $(1)$ and Case $(2)$

$x=t$

$\implies$

$f(x)=x$ $\;$ $\forall x \in \mathbb R$

My doubt: Is my conclusion $f(x)$ is bijective correct just by seeing $f(f(x))=x$?

Am i missing anything?

Other way to solve this Problem is also appreciated

Answer 1

Your conclusion of $f$ been bijective is right, but unnecessary, since you ended proving $f(x)=x$.

You may avoid proving by contradiction to get a more succinct proof:

If $x \le f(x)$, then $f(x) \le f(f(x)) = x$, so $f(x)=x$.

If $x \ge f(x)$, then $f(x) \ge f(f(x)) = x$, so $f(x)=x$.