Continuous Collatz Conjecture Has anyone studied the real function 
$$ f(x) = \frac{ 2 + 7x - ( 2 + 5x )\cos{\pi x}}{4}$$ (and $f(f(x))$ and $f(f(f(x)))$ and so 
on) with respect to the Collatz conjecture? 
It does what Collatz does on integers, and is defined smoothly on all 
the reals. 
I looked at $$\frac{ \overbrace{ f(f(\cdots(f(x)))) }^{\text{$n$ times}} }{x}$$ briefly, and it appears to have bounds independent of $n$. 
Of course, the function is very wiggly, so Mathematica's graph is 
probably not $100\%$ accurate. 

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.
