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ImageByCalculus

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Description

Summary

The image of a set of reals under a quotient of polynomials, { f(x) : x in S }, read
off the function's critical points and its limits: Sullivan and Mackey's §7.3.5 Try 1,
x/(1 + x) over RR \ {-1}, is RR \ {1}, and Prob 7.8.8's
(2x - 1)/(2x (1 - x)) over (0, 1) is RR.

Remarks

A continuous function maps an interval onto an interval, and a differentiable one is
monotone between the zeros of its derivative, so the image of an interval without a pole is
the interval between the least and the greatest of the values at the critical points inside
it, the values at its closed ends and the one-sided limits at its open ones. An end of the
image is closed where the value is taken, at a critical point or a closed end, and open
where it is only approached. The set is cut into such intervals first: at the points a
difference removes, and at the poles, where the function is not defined.
 The derivative's zeros have to be all of them, or a turning point is missed and the image
comes out too small. So the numerator of the derivative, and the denominator whose zeros
are the poles, have to be polynomials of degree four or less, which the solver answers in
radicals completely; the coefficients have to be rational, so that a coefficient which
cancels is zero rather than a residue; and every root has to evaluate, a real one to a
candidate and a non-real one to nothing. A real root can come out of the radicals with an
imaginary residue near 1e-99, which is read as zero below the downcasting tolerance
whether downcasting is on or not.
 https://github.com/asc-community/AngouriMath/issues/1409

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