# Re: [LEAPSECS] The real problem with leap seconds

From: David Malone <dwmalone_at_maths.tcd.ie>
Date: Wed, 11 Jan 2006 12:23:58 +0000

[A lot of discussion on this list seem to revolve around people
understanding terms in different ways. In an impractical example
of that spirit...]

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

To say if TAI is a monotone function of UTC, you need to put an
order on the set of possible TAI and UTC values. To say if UTC is
a continious function of TAI, you need to put a topology on both.

To me, TAI seems to be a union of copies of [0,1) labelled by
YEAR-MM-DD HH:MM:SS where you glue the ends together in the obvious
way and SS runs from 00-59. You then put the obvious order on it
that makes it look like the real numbers.

OTOH, UTC seems to be a union of copies of [0,1) labelled by
YEAR-MM-DD HH:MM:SS where SS runs from 00-60. You glue both the end
of second 59 and 60 to the start of the next minute, in adition to
the obvious glueing.

I haven't checked all the details, but seems to me that you can put
a reasonable topology and order on the set of UTC values that
will make UTC a continious monotone function of TAI. The topology
is unlikely to be Hausdorf, but you can't have everything.

> DTAI jumped
> from 32 s to 33 s; thus, UTC is not a monotone increasing function of
> TAI either.

Since DTAI involves subtracting quantities that aren't real numbers,
you can't conclude that a discontinuity in DTAI results in a
discontinuity in UTC.

David.
Received on Wed Jan 11 2006 - 04:24:18 PST

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