Polygonal length and rectifiability of complex paths

For $a\le b$, a complex path is a map $\gamma :[a,b]\to \mathbb{C}$ continuous from the usual real metric to $d_{\mathbb{C}}(z,w)=\mathrm{\mid }z-w\mathrm{\mid }$ (Intervals of $\mathbb{R}$: the nine order-convex forms, nondegeneracy, and length, The absolute value makes $\mathbb{R}$ a metric space: $d(x,y)=\mathrm{\mid }x-y\mathrm{\mid }$ is a metric, its open balls are the intervals $(x-r,x+r)$, and it is unbounded, Continuity of a map between metric spaces, at a point and globally, in the $\epsilon$-$\delta$ form, The Euclidean metric, convergence, Cauchy sequences, and continuity on the complex plane). Its domain and parametrization are part of the path, not merely its trace.

If $a<b$ and $P=(n,t)$ is a partition of $[a,b]$ as in Partition of $[a,b]$ as a finite strictly increasing list $a=t_0<t_1<\cdots <t_n=b$, its subintervals and their lengths, its mesh, refinement, and the common refinement of two partitions, define

${\mathrm{\ell }}_P^{\mathbb{C}}(\gamma ):={\sum }_{j<n}\mathrm{\mid }\gamma (t_{j+1})-\gamma (t_j)\mathrm{\mid },L_{\mathbb{C}}(\gamma ):={\sup }_P{\mathrm{\ell }}_P^{\mathbb{C}}(\gamma )\in \bar{\mathbb{R}}\mathrm{.}$

The sum is the ordinary finite real sum of the moduli of the chords. Its supremum is taken in the extended real line (The extended real line $\bar{\mathbb{R}}=\mathbb{R}\cup \{-\mathrm{\infty },+\mathrm{\infty }\}$, its order, and the arithmetic that is left undefined, Every subset of $\bar{\mathbb{R}}$ has a least upper bound and a greatest lower bound in $\bar{\mathbb{R}}$, agreeing with the real supremum and infimum on nonempty sets bounded in $\mathbb{R}$). The path is rectifiable when these sums are bounded above by a real number, equivalently when $L_{\mathbb{C}}(\gamma )<+\mathrm{\infty }$. A bounded trace alone does not imply rectifiability; the companion worked example gives a continuous path with bounded trace and unbounded partition sums.

On a singleton interval $[a,a]$, define $L_{\mathbb{C}}(\gamma )=0$ and call every path rectifiable. This convention does not invoke a partition, whose definition requires distinct endpoints.