Suppose $f:[0,1]\to\mathbb R^d$ is continuous, and $P=[t_0,t_1,t_2,\cdots,t_{n-1},t_n]$ is a partition, so $0=t_0<t_1<\cdots<t_n=1$. Define
$$\sum_P\lVert df\rVert=\sum_{1\leq i\leq n}\lVert f(t_i)-f(t_{i-1})\rVert,$$ $$|P|=\max_{1\leq i\leq n}(t_i-t_{i-1}),$$ $$|f(P)|=\max_{1\leq i\leq n}\lVert f(t_i)-f(t_{i-1})\rVert.$$
We consider three definitions of the length of the curve $f$ (the letter $D$ is for "domain", and $C$ for "codomain"):
$$S=\sup_P\sum_P\lVert df\rVert,$$ $$D=\lim_{|P|\to0}\sum_P\lVert df\rVert,$$ $$C=\lim_{|f(P)|\to0}\sum_P\lVert df\rVert.$$
The previous question shows that $S=D$. (In fact it's true even when one of them is infinite.) Now I want to know whether $D=C$.
By uniform continuity of $f$, for any $\varepsilon>0$ there exists $\delta>0$ such that, for all partitions $P$, if $|P|\leq\delta$ then $|f(P)|\leq\varepsilon$.
Suppose $C$ exists. So for any $\eta>0$ there exists $\varepsilon>0$ such that, for all partitions $P$, if $|f(P)|\leq\varepsilon$ then $|\sum_P\lVert df\rVert-C|\leq\eta$. Combining this with uniform continuity, we find that $D$ exists and $D=C$.
Similarly, if $C$ is infinite then $D$ is infinite.
Does this exhaust all the possibilities? Must $C$ exist in $\mathbb R\cup\{\infty\}$?
Suppose the curve is closed, $f(0)=f(1)$, but not constant (not just a point), so its length $S=D$ is positive. Considering the trivial partition $P=[0,1]$, both $0$ and $D$ are limit points of the sums (as $|f(P)|\to0$), so $C$ doesn't exist. To avoid this, let's assume that $f$ is injective.
