On 2006-01-10, Mark Calabretta wrote:

>  I can't let this one pass - UTC is continuous and monotonic.  In fact,
>  ignoring differences in origin, UTC = TAI.  Surprised?  If so then
>  you're confusing a quantity with its representation (though in good
>  company in doing so).

   I do not understand. As a function of TAI, UTC is neither continuous
   nor monotone increasing in the mathematical sense.

   In the defining document, [ITU-R TF.460-6], UTC is given as
                UTC = TAI - DTAI
   where DTAI is a step function of TAI assuming as values only integral
   multiples of 1 s. Unless constant, such a function is not continuous
   on a connected domain in the mathematical sense. Hence, UTC is not
   a continuous function of TAI.

   At some instant when TAI took a value in the positive leap second between
   2006-01-01 + 00 h + 00 min + 32 s and 2006-01-01 + 00 h + 00 min + 33 s
   (the exact instant is not clear from [ITU-R TF.460-6 2002]), DTAI jumped
   from 32 s to 33 s; thus, UTC is not a monotone increasing function of
   TAI either.

   Perhaps you consider UTC as a function of something other than TAI where
   it may well be continuous or monotone (eg, as a function of itself, UTC
   trivially is both continuous and monotone).

            [ITU-R TF.460-6] "Recommendation ITU-R TF.460-6
                             Standard-frequency and time-signal
                             emissions". 2002 Geneva.

   Michael Deckers

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