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I was looking at a situation where $v^{-1}(t) = v(t)$ and then $t = v(v(t))$. I started looking for solutions where the derivative $v'(t)$ is positive. The identity function $v(t) = t$ is a solution, but I didn't find this satisfying for my thinking.

So out of pure interest, I started looking for functions that were both involutory and had positive derivatives. I could not find any other than $v(t) = t$, so I am wondering if it is possible to prove that there are no others, or otherwise provide another example.

Anyone have any ideas?

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Suppose $v$ is such a function. Note that if $v(t) > t$, then $v(v(t)) = t < v(t)$. But then $v$ can't be increasing on any interval containing $t$ and $v(t)$. Similarly if $v(t) < t$.

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