  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.
