If I convert 47°F to Celsius, rounding to the nearest integer, I get 8°C. If I then convert back to Fahrenheit, again rounding, I get 46°F. Back to Celsius, 8°C. Now of course if I continue this process it will remain stable, going back and forth between 8°C and 46°F. Will it always stabilize, for any given starting value?
More generally, suppose I have an arbitrary linear function, $f(x) = ax + b$, with inverse $f^{-1}(x) = \frac {x - b} a$. Using the rounding function $r(x) = \lfloor x + \frac 1 2 \rfloor$, define the round-trip function $g = r \circ f^{-1} \circ r \circ f$. Then the question is, does $g = g \circ g$? If not, is there always some $n$ for which $g^n = g^{n+1}$?
Empirically, it seems that $g$ is idempotent, but the proof has defied my meagre abilities.
Also, are there any more general things we can say? For instance if $f$ isn't necessarily linear, but strictly monotone, does the process of repeatedly applying, rounding, inverting, and rounding again always converge?
PS: Please feel free to add meaningful tags to this... I'm not sure what would be appropriate for this question.
Edit
Vlad has found a counterexample, so let me amend the definition of rounding to be that if the fractional part is exactly .5, it yields the adjacent even number...