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A Riemann–Stieltjes integrable integrand need not be bounded
Example
On , set and for , and let be the unit step at . Then is unbounded but
Facts & Assumptions
Given: The displayed and step integrator .
Only the partition interval across the jump of has a nonzero Stieltjes weight (Riemann–Stieltjes sums, upper and lower sums, and the Riemann–Stieltjes integral).
The function is continuous at (Continuity of at a point of and on : the - condition, its agreement with at a limit point, and continuity at an isolated point).
Verification
For each positive integer , , so the range of is not bounded above.
Every Stieltjes sum equals for a tag in the interval across ; as the mesh tends to zero, . By [L2], these sums tend to . The unbounded behavior near zero is multiplied only by zero increments of .
Depends on
- Riemann–Stieltjes sums, upper and lower sums, and the Riemann–Stieltjes integral
- A one-jump integrator evaluates a continuous integrand at the jump
- Integer powers $a^m$
- 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
- Lower bound, bounded below, bounded set
- 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
Nothing in the library uses this result yet.
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Sources
- W. Rudin, Principles of Mathematical Analysis, Ch. 6, discussion after Definition 6.1 (standard reference, not scraped)