Functional Square Root

[Catullus]

On the topic of function composition. Is the successor function the only function f(x) such that f(f(x)) = f(x)+1?

The equation shows f is one Abel function of itself, and thus also the superfunction of itself.

If you assume that f has a singlevalued inverse function, then by the definition f must satisfy:
\[f^t(x)=f^{-1}(f(x)+t)\Leftrightarrow f^{t+1}(x)=f(x)+t\]
by taking t=-1, we simply get
\[f^0(x)=x=f(x)-1\]
this shows f(x)=x+1 is the only solution.

is also a solution.

What about ? What is f?
What about ?