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$γ (t) = (t, ∣ t ∣)$ on $[- 1, 1]$ is rectifiable of length $2 \sqrt{2}$ but is not differentiable at $0$

Example

The path

$γ : [- 1, 1] \to R^{2}, γ (t) = (t, ∣ t ∣),$

is rectifiable with length $2 \sqrt{2}$ , but it is not differentiable at $0$ and hence is not $C^{1}$ .

Facts & Assumptions

Given: The V-shaped path $γ$ .

[L1]

Vector differentiation is componentwise (The derivative and the Riemann integral of a vector-valued function: an intrinsic derivative and a componentwise integral).

[L2]

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

[L3]

$\sqrt{2}$ is the nonnegative number whose square is $2$ (Square roots exist: a unique $\sqrt{a} \geq 0$ with $(\sqrt{a})^{2} = a$ ; the positives are ${x^{2} : x \neq 0}$ ).

Verification

technique · computation

1.1

On $[- 1, 0]$ the derivative is $(1, - 1)$ , and on $[0, 1]$ it is $(1, 1)$ ; each has Euclidean norm $\sqrt{2}$ by [L3].

givenL1L3

1.2

The right difference quotients at $0$ equal $(1, 1)$ and the left difference quotients equal $(1, - 1)$ . They have different limits, so the vector derivative at $0$ does not exist.

givenL1

2.1

By [L2], the two pieces have lengths $\sqrt{2}$ and $\sqrt{2}$ , so the total length is $2 \sqrt{2}$ .

step 1.1L2

3.1

Thus the path is rectifiable by step 2.1 but not differentiable, and therefore not continuously differentiable, at its corner.

step 2.1step 1.2∎

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · next 3 levels

Direct dependencies and their dependencies through the next three levels: 128 results over 26 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

Rectifiable curve (standard reference, not scraped)

γ(t)=(t,∣t∣)γ(t)=(t,∣t∣) on [−1,1][−1,1] is rectifiable of length 2222​ but is not differentiable at 00