Functions $f$ having the intermediate value property such that $(f(x))^2=x^2$ for all $x\in \mathbb{R}$

Question

It seems "obvious" that there are only four such functions, namely $x,-x,|x|,-|x|$, but I cannot see the rigorous justification for that.

Answer 1

Any other function must have two points $x,y$ with $f(x)=x$, $f(y)=-y$, and $xy\gt 0$. Now apply the intermediate value property to find a point $t$ between $x$ and $y$ with $f(t)=0$; since $t$ can't be $0$ ($x$ and $y$ are both on the same side of $0$), that's your contradiction.