The Real 3x+1 Problem
  In this work, we introduce another extension U of the 3n+1 function to the real line. We propose a conjecture about the U-trajectories that generalizes the famous 3n+1 (or Collatz) conjecture. We then prove our main result about the iterates of U (which is directly related to both of these conjectures). We also introduce the "flipped 3x+1" function \widetilde U and prove an analogous result for its trajectories. In the final section, we pose some interesting questions about the iterates of U (and \widetilde U), prove a couple of simple results about the iterates of U and \widetilde U, introduce other related functions and propose yet more conjectures and questions about their iterates. It's our hope that the results, conjectures and questions presented here will be not only relevant to the 3n+1 conjecture itself, but also of interest in their own right. 
Pavlos B. Konstadinidis