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This question already has an answer here:

Any given setting for $f$ is acceptable.

Iterated function

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marked as duplicate by Did, José Carlos Santos, Martin Sleziak, Community Aug 26 '17 at 10:21

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Apparently I gave a complete solution for this exact problem some years ago, it is one of the links below.

I have the book Kuczma, M., Choczewski B., and Ger, R. (1990). Iterative Functional Equations. Cambridge University Press which was helpful.

The bad news is that the derivative at the fixpoint $x=0$ is $1.$ This means that you can construct Ecalle's solution for $x \geq 0$ so that $f(0) = 0,$ $f'(0) = 1,$ we get $f \in C^\infty,$ and then, for $x > 0,$ we get $x \in C^\omega.$ I should emphasize that there is no holomorphic solution in a neighborhood that includes the origin.

My impression is that we lose smoothness at $x = - \frac{1}{2}.$

Nothing about this is easy.

http://mathoverflow.net/questions/45608/formal-power-series-convergence/46765#46765

http://math.stackexchange.com/questions/208996/half-iterate-of-x2c/209653#209653

http://math.stackexchange.com/questions/911818/how-to-obtain-fx-if-it-is-known-that-ffx-x2x/912324#912324

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