TheoremStatement: Literature-sourcedProof: AI-adaptedSession-authored (Fable 5 assisted)precheck passaudited 2026-08-13

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Arc length is additive across every subdivision point and decreases under restriction

Statement

Let $γ : [a, b] \to R^{n}$ be a path, with $n \geq 1$ , and let $c \in [a, b]$ . Then, in the nonnegative extended reals,

$L_{[a, b]} (γ) = L_{[a, c]} (γ ∣_{[a, c]}) + L_{[c, b]} (γ ∣_{[c, b]}) .$

Consequently $γ$ is rectifiable on $[a, b]$ if and only if both restrictions are rectifiable. The formula includes $c = a$ and $c = b$ through the singleton convention.

Facts & Assumptions

Given: The path and subdivision point $c$ .

[L1]

Inserting a point into a partition does not decrease polygonal length (Refining a partition cannot decrease its inscribed polygonal length).

[L2]

Partitions of adjacent intervals can be concatenated after deleting the repeated common endpoint (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).

[L3]

Arc length is the supremum of polygonal lengths, with singleton length zero (Paths in $R^{n}$ , inscribed polygonal sums, arc length as their supremum, and rectifiability).

Proof

technique · two inequalities

1.1

Concatenating a partition of $[a, c]$ with one of $[c, b]$ gives a partition of $[a, b]$ whose polygonal length is the sum of the two polygonal lengths.

givenL2L3

1.2

Given a partition $P$ of $[a, b]$ , insert $c$ if necessary. By [L1] the refined length is at least $ℓ_{P}$ , and splitting the refined sum at $c$ makes it at most $L_{[a, c]} + L_{[c, b]}$ .

givenL1L2L3

2.1

Taking independent suprema in step 1.1 gives $L_{[a, c]} + L_{[c, b]} \leq L_{[a, b]}$ ; if either left summand is infinite, this already forces the total length to be infinite.

step 1.1L3

3.1

Taking the supremum over $P$ gives the reverse inequality. Together with step 2.1 this proves equality.

step 1.2step 2.1L3

4.1

If $c$ is an endpoint, one summand is zero by [L3]. The equality also shows that the total is finite exactly when both summands are finite.

step 3.1L3∎

Depends on

Used by

Dependency tree · next 3 levels

Direct dependencies and their dependencies through the next three levels: 81 results over 19 levels. An arrow runs from a result to what uses it, and this result sits at the bottom with a heavier outline. Click the chart to enlarge it.

Sources

T. M. Apostol, Mathematical Analysis, Section 6.10 (standard reference, not scraped)