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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.

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Do you assume continuity? – M.B. Oct 27 '12 at 15:38
@M.B. No, continuity is not necessary – Golbez Oct 27 '12 at 15:43
up vote 1 down vote accepted

This is not an answer, but can lead to one. Here are a few remarks on your problem:

  • since $f \circ g$ and $g \circ f$ are both injective it follows that $f,g$ are injective functions, which coupled with the fact that $f,g$ are onto prove that $f,g$ are bijections of $\Bbb{R}$ in $\Bbb{R}$.

  • note that $f,g$ cannot be monotonic on the same interval, because then $f\circ g$ and $g \circ f$ would have the same monotonicity on that interval.

  • an injective, continuous function is monotonic. Therefore $f,g$ cannot be both continuous on an interval $I$.

These being said, if there are some examples of functions like in your problem, then they will look very nasty. If one is continuous on an interval $I$ then the other one will not be continuous on any subinterval of $I$.

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Thanks for your hint! – Golbez Oct 27 '12 at 15:43

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