Rectifiable curve

Definition

Let $\gamma \colon [a, b] \to \mathbb{R}^n$ be a parametrised curve. For a partition $P$ given by $a = t_0 < t_1 < \dots < t_k = b$, let $$L(\gamma, P) = \sum_{i = 1}^{k} \|\gamma(t_i) - \gamma(t_{i-1})\|$$ be the length of the inscribed polygonal path. The curve $\gamma$ is rectifiable if $\sup_P L(\gamma, P) < \infty$, where the supremum is over all partitions of $[a, b]$. Then the length of $\gamma$ is $\ell(\gamma) = \sup_P L(\gamma, P)$.

Remarks

If $\gamma$ is of class $C^1$, then it is rectifiable and $\ell(\gamma) = \int_a^b \|\gamma'(t)\| \, dt$. A refinement of $P$ does not decrease $L(\gamma, P)$, by the triangle inequality. The graph of $f(x) = x \sin(1/x)$ for $0 < x \le 1$, $f(0) = 0$, is a continuous curve that is not rectifiable. The Koch curve is another example. A curve is rectifiable if and only if each component $\gamma_i$ has bounded variation on $[a, b]$.

Explanations

L1 human

Approximate the curve by a chain of straight segments between points on it, and add up their lengths. Finer chains give longer totals. If the totals stay bounded, the curve has a finite length, and the length is their least upper bound.