[17] in mathematical software users group
Re: A formalism for discontinuities of functions?
daemon@ATHENA.MIT.EDU (daemon@ATHENA.MIT.EDU)
Mon Mar 23 11:11:41 1992
To: msug@Athena.MIT.EDU
Date: Mon, 23 Mar 92 11:10:45 EST
From: Reid M. Pinchback <reidmp@Athena.MIT.EDU>
chen@deakin.OZ.AU writes:
>SymbMath can deal with discontinuous function by x+zero for right-hand side
>limit value or by x-zero for left-hand side limit value. This type of
>functions can be evaluated, differentiated, even integrated. e.g.
>
>- ->f(x)=1/x # define function
>- ->f(0+zero)
>inf # right-hand side limit value (from +inf to 0)
>- ->f(0-zero)
>- -inf # left-hand side limit value ( from -inf to 0)
>- ->exp(1/(0+zero))
>inf
>- ->exp(1/(0-zero))
>0
Yes, these are the kind of situations that Maple can deal with too, and
I suspect the same would be the case for Mathematica. But this is very
different from the situation that I was describing. Here, you are
avoiding the discontinuity at a *KNOWN* point. This doesn't generalize
to non-trivial situations at all. Try working out the locations of
discontinuities for rational composite trig functions, like:
cos(1 + x * arcsin(x))
----------------------
sin(exp(1/x) * cos(x))
Now, to plot this as a smooth curve, you couldn't use cubic splines.
To integrate it, although you could try and "grind" it through a process
of symbolic integration, I don't think you'd get very far. To evaluate
it numerically, you need to figure out over what interval you want the
integration performed... and you are back to needing to know the locations
(and some other qualitative information) of the discontinuities again.
This example is relatively "easy"... things probably get even worse when you
start to work with measure functions.
It is easy for limits to "deal" with discontinuities... because they don't
really have to. Continuity is defined in terms of limits, not the other
way around, so we use limits (or lack thereof) to test discontinuity.
The construction of integrability is similar, so again sometimes the issue
is avoided (eg: in a close interval, you can have a countable number of
discontinuities where the left- and right-hand limits exist, and this does
not impact integrability).
Differentiability, on the other hand, requires continuity, so you need to
know where those points are in order to "avoid" them. This is why splines
"break" for discontinuous functions... the technique assumes the existance
of several orders of continuous derivates.
What is really needed is some way to restrict the domain of a function, f,
to avoid the discontinuities... which results in a partition of the
domain. Two points would be equivalent iff they could be "placed" in
the same unbroken interval. The resulting intervals would be the
equivalence classes of the partitioned domain of f. You could then speak
of the restriction of f to an equivalence class, and apply cubic spline
techniques to each such restriction. If the number of classes is finite
(or maybe even countable), then you may have a general "algorithm" for
"dealing" with non-continuous functions.
Reid M. Pinchback
Faculty Liaison
Academic Computing Services, MIT
