〈 Berger | Dillon 〉

I wonder what a function f(x) with the property that f(f(x)) = e^x might look like. (My usual caveat: DON’T TELL ME & RUIN THE WONDERING STAGE.)

I’ll solve this numerically tomorrow and see if it looks like anything I recognize, and/or if curve fitting to the solution gives something correct. I don’t see any reason f can’t be holomorphic though, and I’m pretty sure Picard-Lindelöf gives us existence and uniqueness

In particular, the derivative is never zero and everywhere continuous, so we also have a guarantee that f is invertible (by whatever that theorem is called that says a function is locally invertible if its derivative is continuous and nonzero in that neighborhood)

But doesn’t that just say that f(f(x)) is is invertible? Pardon my analysis is a bit rusty

From the expression you have above in the box, if you assume f is smooth enough and defined for all R then f' can never vanish. And then it's inverse is found by just drawing the graph and reflecting on the x=y line (I mean there are theorems, but that's even easier :) )

Of course, the assumption that it is well-defined on all of R needs to be justified