Bounded variation and total variation on an interval

Depends on

• 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
• Finite sums and finite products, by recursion
• Laws of finite sums and finite products
• Lower bound, bounded below, bounded set
• Complete ordered field (least-upper-bound property)
• Suprema and infima are unique
• Intervals of $\mathbb{R}$: the nine order-convex forms, nondegeneracy, and length
• Absolute value in an ordered field

Used by

• A bounded-variation integrand is Riemann–Stieltjes integrable against every continuous integrator Corollary
• Brownian paths have infinite total variation Corollary
• The total-variation bound for a Riemann–Stieltjes integral Corollary
• A common jump can destroy Riemann–Stieltjes integrability Counterexample
• A continuous function on [0,1] can have unbounded variation Counterexample
• Bounded-variation Riemann-Stieltjes theory does not construct the Brownian Ito integral Counterexample
• The continuous path γ(x)=(x,x sin(1/x)) on [0,1], with γ(0)=(0,0), is not rectifiable Counterexample
• Quadratic covariation of Brownian Ito processes Definition
• Total-variation function on a compact interval Definition
• Variation function and positive and negative variations Definition
• A BV function can fail continuity at one point and still be differentiable almost everywhere Example
• The Cantor function is continuous and of bounded variation but not absolutely continuous Example
• The graph path t↦(t,c(t)) of the Cantor function is rectifiable although its second coordinate is not absolutely continuous Example
• Young's theorem integrates a Hölder function of unbounded variation against itself Example
• FALSE: every continuous function on a compact interval has a rectifiable graph False statement
• FALSE: every function of bounded variation is absolutely continuous False statement
• Every bounded-variation function is uniformly approximable by step functions Lemma
• Homogeneity and subadditivity of total variation Lemma
• Refinement and tag-change estimates for Stieltjes sums Lemma
• Total variation bounds increments; bounded-variation functions are bounded; zero variation means constant Lemma
• Total variation is additive over adjacent subintervals and decreases under restriction Lemma
• Conventions and proved scope for bounded variation and Stieltjes integration Remark
• A bounded function with finitely many discontinuities is Stieltjes integrable against a continuous bounded-variation integrator Theorem
• A continuous integrand is Riemann–Stieltjes integrable against every bounded-variation integrator Theorem
• A countable pure-step integrator evaluates a continuous integrand as the absolutely convergent weighted sum of its values at the jumps Theorem
• A path in ℝⁿ is rectifiable exactly when every coordinate has bounded variation Theorem
• Banach--Zarecki characterisation of absolute continuity Theorem
• C¹ implies Lipschitz, Lipschitz implies absolutely continuous, and absolutely continuous implies continuous and bounded variation Theorem
• Every function of bounded variation is differentiable almost everywhere Theorem
• Functions of bounded variation form an algebra Theorem
• Jordan decomposition for functions of bounded variation Theorem
• Linearity and interval additivity of the Riemann–Stieltjes integral Theorem
• Two bounded-variation functions with no common discontinuity are Riemann–Stieltjes integrable Theorem

Dependency tree · two levels

28 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

• William F. Trench, Introduction to Real Analysis, Ch. 3 (standard reference, not scraped)
• Christopher Heil, Absolute Continuity and the Banach-Zaretsky Theorem (standard reference, not scraped)