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A one-jump integrator evaluates a continuous integrand at the jump
Example
Fix and let for and for . For every continuous ,
Facts & Assumptions
Given: The one-jump integrator and a continuous .
A Stieltjes sum weights each tag by the corresponding integrator increment (Riemann–Stieltjes sums, upper and lower sums, and the Riemann–Stieltjes integral).
Verification
In every tagged partition, all increments of vanish except the one on the interval across its jump; that increment is one. Its tag satisfies , so the complete Stieltjes sum is and tends to by [L2].
The same computation counts a jump at . A jump at is represented by and for and likewise contributes . By contrast, merely assigning a constant endpoint value on the whole interval creates no increment and hence has integral zero.
Depends on
- Riemann–Stieltjes sums, upper and lower sums, and the Riemann–Stieltjes integral
- A continuous integrand is Riemann–Stieltjes integrable against every bounded-variation integrator
- Continuity of $f : A \to \mathbb{R}$ at a point of $A$ and on $A$: the $\varepsilon$-$\delta$ condition, its agreement with $\lim_{x \to c} f(x) = f(c)$ at a limit point, and continuity at an isolated point
- Partition of $[a,b]$ as a finite strictly increasing list $a = t_0 < t_1 < \dots < t_n = b$, its subintervals and their lengths, its mesh, refinement, and the common refinement of two partitions
Used by
Dependency tree · two levels
36 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.
Sources
- W. Rudin, Principles of Mathematical Analysis, Ch. 6, Theorem 6.15 (standard reference, not scraped)