Does there exists $f: \mathbb{R} \to \mathbb{R},g: \mathbb{R} \to \mathbb{R}$, such that $f,g$ are onto function and satisfies:
$f(g(x))$ strictly monotonically increasing and $g(f(x))$ strictly monotonically decreasing.
This question occur when I realize that if $g,f$ are monotonic functions with same monotonicity, $g(f(x))$ will increase, if they are monotonic functions with opposite monotonicity, $g(f(x))$ will decrease. But what about $g,f$ are not monotonic?
I haven't got any idea about it, thanks alot for your help.