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BreakpointIntegration

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Description

Summary

A definite integral whose integrand can jump inside the range, split at the jumps. A
piecewise breaks where a case's condition changes truth; floor(x) and
ceil(x) break at every whole number. Between two breakpoints the piecewise is one
of its cases and the step is one constant, so the pieces integrate through an
antiderivative and add.

Remarks

F(b) - F(a) is the integral only where F is continuous on [a, b], and
an antiderivative found with the piecewise's cases or the floor taken as constants is
continuous between two jumps and not across one. Taken across one it answered
integral(x - floor(x), x, 0, 3) with 0 and the tent map's integral over
[0, 1] with 1, where they are 3/2 and 1/2; with a symbolic
bound it answered (n - floor(n))^3 / 3, and with a piecewise it handed back a
piecewise that still mentioned the integration variable. None of that happens here: an
integrand with a break never goes through an antiderivative between two bounds.
Finitely many breakpoints -- a piecewise whose conditions compare the variable with
numbers -- are collected, the range is cut at the ones inside it, and on each piece the
piecewise is replaced by the case that holds at the piece's midpoint, which is exact
because the condition has no other place to change. A condition that compares the
variable with something symbolic, or a bound that is symbolic, is declined: the pieces
depend on where the jumps fall, and answering across them is what this exists to stop.
A piecewise whose conditions do not mention the variable is a constant here and goes
through the antiderivative as one.
Infinitely many, evenly spaced -- a floor or a ceiling of the variable -- are the
same split written as a sum: on [n, n + 1) the floor is n, the ceiling
n + 1, x is n + t with t over [0, 1), and the integral
over whole bounds is a sum over n of an integral over t with no step in it,
which the summation's closed forms answer; +oo is allowed as the upper bound. A
numeric bound that is not whole contributes the piece up to the nearest whole number,
on which the step is one constant. The fractional part x - floor(x) is substituted
as a unit, since written out it arrives as n + t - n, and a term subtracted from
itself simplifies to a conditional zero rather than to nothing
(#1174).
 Offered only where every piece then resolves; an integral rewritten as an unevaluated sum
of unevaluated integrals is not an answer. Question I.2 of
#1212, and the
review of #1215,
which asked for the general mechanism rather than the floor's special case.

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