Conventions and proved scope for bounded variation and Stieltjes integration

Total variation is zero on a singleton, and both ordinary and Riemann–Stieltjes integrals use the oriented convention when endpoints are reversed. Absolute continuity here is formulated with finite disjoint families of intervals.

On a nondegenerate interval $[a,b]$ with $a<b$, and for a nondecreasing integrator, the weighted Darboux condition matches the all-fine-mesh definition only together with continuity of the integrand at the integrator's discontinuities; this extra compatibility is vacuous for a continuous integrator. The hypothesis $a<b$ is part of the statement and not cosmetic: on $[a,a]$ the integral is $0$ by convention, so every bounded integrand is integrable there, while a singleton interval admits no partition at all and so the Darboux condition fails; a consumer needing $a=b$ reads the value off the definition instead. A general BV integrator is handled through Jordan decomposition or tagged sums. Finite-step integrators turn the integral of a continuous integrand into a weighted evaluation sum over the jumps, while continuously differentiable integrators reduce the integral of a Riemann-integrable integrand to an ordinary integral against the derivative. The no-common-discontinuity theorem is sharp in view of the companion common-jump counterexample. Young's theorem is proved here only for rational Hölder exponents because arbitrary real exponents are not available at this point in the reading order; the later Real powers for positive bases, with the zero-base positive-exponent convention ↗ is what supplies them. No Lebesgue–Stieltjes measure, almost-everywhere differentiability theorem, or arbitrary-real-exponent Stieltjes theorem is asserted on this page.