> From: Raul Miller <[EMAIL PROTECTED]> > To: Programming forum <[email protected]> > Sent: Friday, September 5, 2008 6:29:07 PM > Subject: Re: [Jprogramming] deriving discontinuous functions from polynomials > > On Fri, Sep 5, 2008 at 6:16 PM, John Randall > wrote: > > Are you looking for the implicit function theorem? This gives a local > > inverse on the graph of an implicit function whenever the tangent line > > is not horizontal. > > This this theorem looks very relevant. > > But I will have to spend some time digesting it.
There is a simpler "Inverse function theorem" http://planetmath.org/encyclopedia/InverseFunctionTheorem.html The idea is simply that for y=f(x), on a monotonous stretch where f(') != 0 and f in C^1 (continuously differentiable), then there exists x=g(y) on image of f on the stretch. It doesn't tell you how to find the stretches or inverses. For inverses, the stretches are probably between potential extrema, ie f(')(x)=0. In inverse, where the graph is flipped, the stretches can overlap, but overlaps can be detected looking at segments from left to right, since original f was a map. Thus we get non-overlapping stretches. Each such stretch can be approximated with a polynomial from a sample of points obtained with f. Does that sound like a recipe for inverse function? ---------------------------------------------------------------------- For information about J forums see http://www.jsoftware.com/forums.htm
