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Sawtooth paths converge uniformly to a line segment while every sawtooth has length $\sqrt{2}$ and the limit has length $1$

Counterexample

For each integer $k \geq 1$ , let $γ_{k} : [0, 1] \to R^{2}$ be the polygonal path through

$p_{k, j} = (\frac{j}{2 k}, \frac{1 - (- 1)^{j}}{4 k}), 0 \leq j \leq 2 k,$

at the corresponding parameters $j / (2 k)$ . Then $γ_{k}$ converges uniformly to $γ (t) = (t, 0)$ , but

$L (γ_{k}) = \sqrt{2} for every k, L (γ) = 1.$

Facts & Assumptions

Given: The zigzag paths above.

[L1]

A continuous piecewise- $C^{1}$ path has length equal to the sum of the integrals of its speeds over the pieces (A continuous piecewise- $C^{1}$ path is rectifiable and its length is the sum of the speed integrals over its pieces).

[L2]

The integral of a constant $C$ on $[u, v]$ is $C (v - u)$ (If $m \leq f \leq M$ on $[a, b]$ then $m (b - a) \leq L (f, P) \leq \underline{\int_{a}^{b}} f \leq \overline{\int_{a}^{b}} f \leq U (f, P) \leq M (b - a)$ for every partition $P$ ; in particular every constant function is integrable, with $\int_{a}^{b} c = c (b - a)$ ).

[L3]

The sequence $1 / k$ of real numbers tends to zero (For every $ε > 0$ in a complete ordered field there is a natural $n \geq 1$ with $1 / n < ε$ ).

[L4]

Uniform convergence guarantees only $L (γ) \leq lim inf L (γ_{k})$ (Arc length is lower semicontinuous under uniform convergence of paths).

Verification

technique · construction

1.1

Every $γ_{k} (t)$ has first coordinate $t$ and second coordinate between $0$ and $1 / (2 k)$ , so $\sup_{t} ∥ γ_{k} (t) - γ (t) ∥_{2} \leq 1 / (2 k) \to 0$ by [L3].

givenL3

1.2

On each of the $2 k$ parameter intervals, $γ_{k}$ has derivative $(1, 1)$ or $(1, - 1)$ and hence constant speed $\sqrt{2}$ . By [L1]--[L2], each piece contributes $\sqrt{2} / (2 k)$ and $L (γ_{k}) = \sqrt{2}$ .

givenL1L2algebra

1.3

The limit path has constant derivative $(1, 0)$ and speed $1$ , so [L1]--[L2] give $L (γ) = 1$ .

givenL1L2algebra

2.1

Thus lengths do not converge to the length of the uniform limit. The valid inequality [L4] reads $1 \leq \sqrt{2}$ , as expected.

step 1.1step 1.2step 1.3L4∎

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

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Sources

J. Denzler, Calculus of Variations, Section 4.7 (standard reference, not scraped)

Sawtooth paths converge uniformly to a line segment while every sawtooth has length 22​ and the limit has length 11