Continuous Collatz Conjecture

Solution 1:

The analogue of the Collatz conjecture would be false, since the image of an interval of length $1$, say, contains an interval of length $1$ strictly to the right. That means you can find a point whose images keep moving to the right.

Solution 2:

Yes, it has been studied in

Xing-yuan Wang and Xue-jing Yu (2007), Visualizing generalized 3x+1 function dynamics based on fractal, Applied Mathematics and Computation 188 (2007), no. 1, 234–243. (MR2327110).

I have found this reference in Jeffrey Lagarias's "The $3x + 1$ Problem: An Annotated Bibliography, II (2000-2009)".

This real/complex interpolation of yours is not the only one imaginable, you will find several others either in Lagarias' article cited above, or in the first part of this series of articles, namely in "The $3x+1$ problem: An annotated bibliography (1963--1999)".

Solution 3:

Yes people have studied that, as well as extending it to the complex plane, e.g.:

Chamberland, Marc (1996). "A continuous extension of the 3x + 1 problem to the real line". Dynam. Contin. Discrete Impuls Systems. 2 (4): 495–509.

Letherman, Simon; Schleicher, Dierk; Wood, Reg (1999). "The (3n + 1)-Problem and Holomorphic Dynamics". Experimental Mathematics. 8 (3): 241–252. Cached version at CiteSeerX

You'll also find something about that in the corresponding wikipedia article, including the Collatz fractal.