[13] in mathematical software users group
A formalism for discontinuities of functions?
daemon@ATHENA.MIT.EDU (daemon@ATHENA.MIT.EDU)
Thu Mar 19 13:56:16 1992
To: msug@Athena.MIT.EDU
Date: Thu, 19 Mar 92 13:53:41 EST
From: Reid M. Pinchback <reidmp@Athena.MIT.EDU>
Does anybody know of an existing formalism that characterizes where
functions have discontinuities? I'm primarily interested in univariate
functions, like functions f : R -> R.
I've fiddled with this a bit, and so far the best I've been able to
come up with is to try and create some kind of algebra over functions
and their discontinuities, but so far the results look like they'd only
be reasonable for "easy" situations, and err on the side of finding more
discontinuities than are probably there.
For those that ask... "why would you want this?", consider doing plotting
using cubic splines. This technique assumes that functions are so smooth
you'd think they had teflon coatings. In practise, less smooth functions
are manageable, but discontinuities are definitely unacceptable. This
means that the plots of a lot of reasonable functions like 1/sin(x) don't
come out even close to what they should be. To get a proper plot you have
to (manually) split the function over various subranges and then plot the
set of smaller functions that you then have. If this was only an issue
of plotting it may just seem a minor inconvenience, but since splines
approximate the original function they have other applications (like
approximating integrals).
For dealing with one specific function, manually coming up with a way
to partition the domain around the (known) discontinuities is ok... but
if you want to create a routine to deal arbitrary functions, then this
no longer works.
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Reid M. Pinchback
Faculty Liaison
Academic Computing Services, MIT
