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A continuous function on can have unbounded variation
Example
Let and define Then is continuous on but has unbounded variation.
Facts & Assumptions
Given: The displayed function .
Distance to a nonempty set is -Lipschitz (, so the distance to a fixed nonempty set is -Lipschitz).
The integer-part property gives an integer with ; one of and is at most , so . The definition gives value zero at integers, and the two nearest integers to are both at distance (Integer part: for every real there is exactly one integer with , The integers as equivalence classes of pairs of naturals).
The harmonic series diverges (For rational , converges iff at exponent ).
Verification
By [L1] and the algebra of limits, is continuous for . By [L2], , so as . Hence is continuous on .
For every integer , , while [given] The ordered partition containing these alternating zeros and peaks for therefore has variation sum at least , apart from at most one endpoint term.
Since and the harmonic partial sums are unbounded by [L3], the variation sums in step 1.2 are unbounded. Thus the continuous function is not BV.
Depends on
- Bounded subset, diameter, distance from a point to a set, and distance between two sets in a metric space
- $|d(x,A) - d(y,A)| \le d(x,y)$, so the distance to a fixed nonempty set is $1$-Lipschitz
- The absolute value makes $\mathbb{R}$ a metric space: $d(x,y) = |x-y|$ is a metric, its open balls are the intervals $(x-r, x+r)$, and it is unbounded
- Dictionary: for $A \subseteq \mathbb{R}$ with the metric $d(x,y) = |x-y|$, continuity and uniform continuity of $f : A \to \mathbb{R}$ agree with the metric-space notions, the Lipschitz and Hölder conditions are the metric ones instantiated, and a subset of $\mathbb{R}$ is compact in the open-cover sense of $\mathbb{R}$ exactly when it is a compact metric subspace
- The integers as equivalence classes of pairs of naturals
- Integer part: for every real $x$ there is exactly one integer $m$ with $m \le x < m + 1$
- Sums, scalar multiples, products and quotients of function limits, the quotient under the hypothesis that the denominator limit is nonzero
- 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
- Basic properties of the absolute value
- Bounded variation and total variation on an interval
- 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
- For rational $p > 0$, $\sum 1/k^p$ converges iff $p > 1$
Used by
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Sources
- William F. Trench, Introduction to Real Analysis, Section 3.2 (standard reference, not scraped)