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-if both f and g have the same monotony => fοg is increasing

-if f and g have different monotony => fοg is decreasing

But the reciprocal of this is not always true (easy to prove with a contradicting example).

Though, it came to my mind that, if we have a function h : D -> D, a kind of of reciprocal might be valid for hοh.

I think that if hοh is monotonic it results that h is either decreasing or increasing, but I am not sure if it is true or not, neither how to prove or disprove it. This is actually my question, is it true and how you prove that?