Complex Coordinate and Path Transport

Write $z=a+bi$. The embedded scalar has coordinates $(r,0)$ by [F2], so the coordinate multiplication formula in [F1] gives $\mathrm{\Phi }(rz)=(ra,rb)=r(a,b)=r\mathrm{\Phi }(z)$ for every $r$, including $r=0$.

Additivity and bijectivity are [F1]; step 1.1 supplies real homogeneity. These are exactly the stated real-linearity conditions, so $\mathrm{\Phi }$ is a real-linear bijection.

If $a<b$, every partition chord contributes $\mathrm{\mid }\gamma (t_{j+1})-\gamma (t_j)\mathrm{\mid }=d_{\mathbb{C}}(\gamma (t_{j+1}),\gamma (t_j))$, a finite nonnegative real by [F1]. The finite sum is therefore a finite nonnegative real by [F2].

The set of partition sums is nonempty: the list with $n=1$, $t_0=a$, $t_1=b$, and tail $t_k=b$ is a partition. Thus [F3] supplies its extended-real supremum. Since every sum is nonnegative, the supremum is not $-\mathrm{\infty }$; it is finite exactly when some real number bounds all partition sums above. This is the stated rectifiability criterion.

If $a=b$, the definition assigns length $0$ and rectifiability directly; no partition is available or used.

For $z,w\in \mathbb{C}$, [F2] gives $d_2(\mathrm{\Phi }z,\mathrm{\Phi }w)=d_{\mathbb{C}}(z,w)$, so bijectivity [F1] makes $\mathrm{\Phi }$ an isometry. For every $r>0$, $u\in B_{\mathbb{C}}(z,r)$ exactly when $\mathrm{\Phi }u\in B_2(\mathrm{\Phi }z,r)$; hence $\mathrm{\Phi }[B_{\mathbb{C}}(z,r)]=B_2(\mathrm{\Phi }z,r)$, and the inverse has the reverse correspondence. If $U\subseteq \mathbb{C}$ is open, let ${\mathcal{B}}_U$ be the family of all open balls contained in $U$. Every point of $U$ lies in one such ball by the definition of openness, and every member of ${\mathcal{B}}_U$ is contained in $U$, so $U=\bigcup {\mathcal{B}}_U$. The ball correspondence maps this union to a union of open Euclidean balls; the arbitrary-union clause of [F3] makes $\mathrm{\Phi }[U]$ open. The same argument for the inverse shows that both maps are open. Equality of distances gives continuity of both maps with $\delta =\epsilon$, so the bijection is a homeomorphism by [F3].

Real-linearity [F1] gives $\mathrm{\Phi }((1-t)z+tw)=\mathrm{\Phi }((1-t)z)+\mathrm{\Phi }(tw)=(1-t)\mathrm{\Phi }(z)+t\mathrm{\Phi }(w)$ for every real $t$.

By the ball correspondence in step 1.1, a ball around $z$ is contained in $A$ exactly when its image ball is contained in $\mathrm{\Phi }[A]$; likewise a ball around $z$ meets $A$ exactly when its image ball meets $\mathrm{\Phi }[A]$. The definitions in [F4] therefore give the interior and closure identities. Bijectivity preserves set difference, so it also gives the boundary identity. Openness transfers by the same ball test, and closedness transfers because complements correspond under the bijection.

If $A=\mathrm{\varnothing }$, then $\mathrm{\Phi }[A]=\mathrm{\varnothing }$ and boundedness, compactness, and connectedness agree for the identical empty subsets. Otherwise, ball correspondence transfers containment in a ball in both directions, proving boundedness. The restrictions $\mathrm{\Phi }{\mathrm{\mid }}_A$ and ${\mathrm{\Phi }}^{-1}{\mathrm{\mid }}_{\mathrm{\Phi }[A]}$ are continuous. Applying the compact-image theorem [F6] to these restrictions proves compactness in both directions; applying the connected-image theorem [F7] proves connectedness in both directions.

The inverse ${\mathrm{\Phi }}^{-1}$ is real-linear: for arbitrary $u,v\in {\mathbb{R}}^2$, choose $\zeta ,\omega \in \mathbb{C}$ with $\mathrm{\Phi }(\zeta )=u$ and $\mathrm{\Phi }(\omega )=v$, possible by [F1]. Additivity of $\mathrm{\Phi }$ gives ${\mathrm{\Phi }}^{-1}(u+v)={\mathrm{\Phi }}^{-1}(\mathrm{\Phi }(\zeta +\omega ))=\zeta +\omega ={\mathrm{\Phi }}^{-1}u+{\mathrm{\Phi }}^{-1}v$; for every $\lambda \in \mathbb{R}$, its real homogeneity gives ${\mathrm{\Phi }}^{-1}(\lambda u)={\mathrm{\Phi }}^{-1}(\mathrm{\Phi }(\lambda \zeta ))=\lambda \zeta =\lambda {\mathrm{\Phi }}^{-1}u$.

The correspondence $L\mapsto \mathrm{\Phi }\circ L\circ {\mathrm{\Phi }}^{-1}$ sends each real-linear $L$ to a real-linear map $A$ by [F1], [F2], and step 1.1. Conversely, for each real-linear $A$, the map $L={\mathrm{\Phi }}^{-1}\circ A\circ \mathrm{\Phi }$ is real-linear by the same facts, and the two constructions are inverse because ${\mathrm{\Phi }}^{-1}\circ \mathrm{\Phi }$ and $\mathrm{\Phi }\circ {\mathrm{\Phi }}^{-1}$ are identity maps. Thus this is a bijection between the real-linear maps on the two coordinate spaces.

For corresponding $L,A$ from step 2.1, put $k=\mathrm{\Phi }(h)$. Additivity gives $\mathrm{\Phi }(z+h)=\mathrm{\Phi }(z)+k$. If $r(h)=f(z+h)-f(z)-L(h)$ and $\rho (k)=f_{\mathbb{R}}(\mathrm{\Phi }(z)+k)-f_{\mathbb{R}}(\mathrm{\Phi }(z))-A(k)$, then the definition of $f_{\mathbb{R}}$ and $A(k)=\mathrm{\Phi }(L(h))$ give $\rho (k)=\mathrm{\Phi }(r(h))$. The isometry [F1] therefore gives $\mathrm{\parallel }\rho (k){\mathrm{\parallel }}_2\mathrm{/}\mathrm{\parallel }k{\mathrm{\parallel }}_2=\mathrm{\mid }r(h)\mathrm{\mid }\mathrm{/}\mathrm{\mid }h\mathrm{\mid }$ for $h\mathrm{\ne }0$; the bijective isometry also sends $h\to 0$ exactly to $k\to 0$ and sends the condition $z+h\in U$ exactly to $\mathrm{\Phi }(z)+k\in \mathrm{\Phi }[U]$. Hence the two remainder limits are equivalent.

By [F3], total differentiability of $f_{\mathbb{R}}$ at $\mathrm{\Phi }(z)$ means that some real-linear $A$ has Euclidean remainder ratio tending to zero. Steps 2.1 and 3.1 identify such maps $A$ bijectively with real-linear $L$ and identify their remainder ratios, so this condition is equivalent to existence of the stated $L$ with the complex remainder ratio tending to zero. For each corresponding pair, the Euclidean derivative is $A=\mathrm{\Phi }\circ L\circ {\mathrm{\Phi }}^{-1}$.

The path $\mathrm{\Gamma }$ is continuous: for any $s,t\in [a,b]$, [F1] gives $d_2(\mathrm{\Gamma }(s),\mathrm{\Gamma }(t))=d_{\mathbb{C}}(\gamma (s),\gamma (t))$, so the continuity condition for $\gamma$ implies that for $\mathrm{\Gamma }$. By [F3], $\mathrm{\Gamma }=(x,y)$. For each partition point pair, [F1] identifies the chord lengths; summing termwise gives equal polygonal sums by [F2].

If $a<b$, taking the supremum of those equal sums over the identical partition set in [F2] gives $L_{\mathbb{C}}(\gamma )=L_{{\mathbb{R}}^2}(\mathrm{\Gamma })$. If $a=b$, both lengths are zero by [F2].

Apply [F4] to the continuous ${\mathbb{R}}^2$ path $\mathrm{\Gamma }$. The length equality in step 2.1 identifies its rectifiability with that of $\gamma$, and [F3] identifies its coordinate functions with $x$ and $y$. Thus the bounded-variation equivalence holds in both directions.

Under the piecewise-$C^1$ hypothesis with $a<b$, [F5] applies to $\mathrm{\Gamma }$ on the given subdivision. Substituting the length equality from step 2.1 gives the first integral formula and finiteness; [F6] gives the displayed square-root integrand. No derivative equality at adjacent endpoints is required by [F5].

If $a=b$, [F2] gives zero for both lengths, [F4] gives zero coordinate variations, and the speed sum is empty. Hence all stated boundary conventions hold.

The composite $\gamma \circ \phi$ is continuous. Indeed, fix $s_0\in [c,d]$ and $\epsilon >0$. By continuity of $\gamma$ at $\phi (s_0)$ choose $\delta >0$ such that $u\in [a,b]$ and $\mathrm{\mid }u-\phi (s_0)\mathrm{\mid }<\delta$ imply $d_{\mathbb{C}}(\gamma (u),\gamma (\phi (s_0)))<\epsilon$. By continuity of $\phi$ at $s_0$ choose $\eta >0$ such that $s\in [c,d]$ and $\mathrm{\mid }s-s_0\mathrm{\mid }<\eta$ imply $\mathrm{\mid }\phi (s)-\phi (s_0)\mathrm{\mid }<\delta$. The implication in [F5] then proves continuity of the composite at $s_0$.

If $c=d$, surjectivity forces $a=b$, so both lengths are zero by [F1]. If $a=b$ while $c<d$, then $\gamma$ and $\gamma \circ \phi$ are constant paths and both lengths are zero by [F1]. Thus the equality covers all singleton cases.

Put $\mathrm{\Gamma }=\mathrm{\Phi }\circ \gamma$. By the isometry in [F2], $\mathrm{\Gamma }$ is continuous and $\mathrm{\Phi }\circ (\gamma \circ \phi )=\mathrm{\Gamma }\circ \phi$. The Euclidean reparametrization theorem [F3] gives $L_{{\mathbb{R}}^2}(\mathrm{\Gamma }\circ \phi )=L_{{\mathbb{R}}^2}(\mathrm{\Gamma })$.

Apply the complex-to-coordinate length equality [F2] to $\gamma$ and, using step 1.1, to $\gamma \circ \phi$. Together with step 2.1 this yields $L_{\mathbb{C}}(\gamma \circ \phi )=L_{\mathbb{C}}(\gamma )$.

By [F4], the nondecreasing case preserves order and the nonincreasing case reverses it. If $a<b$, a nondecreasing surjection must attain $a$ at $c$ and $b$ at $d$; a nonincreasing surjection must attain $b$ at $c$ and $a$ at $d$. If $a=b$, the map is constant and there is no nontrivial orientation.