Rectifiable curve has $n$-dimensional Jordan content zero

Question

Question $14.10$, Chapter $14$, in Apostol's "Mathematical Analysis" ($2^{\text{nd}}$ edition):

Let $\Gamma$ be a rectifiable curve in $\mathbb{R}^{n}$. Prove that $\Gamma$ has $n$-dimensional Jordan content zero.

Note that this question differs from How prove that the range of an rectifiable curve has measure zero? and Prove that the image of a curve has zero content in that there is no assumption of continuity here.

I've had two possible ideas, but haven't managed to finish a proof with either one.

The first involves approximating the curves with line segments so that a finite covering by $n$-dimensional intervals of the (finite number of) segments also contains the curve, provided that the approximation is 'close enough'. The problem I run into is approximating the (possibly countably many) disjoint curve segments (as a result of possibly countably many discontinuities) with only a finite number of segments at my disposal.

The second idea would make use of Theorem $6.17$, wherein one learns that for a path $\mathbf{f} \colon [a,b] \to \mathbb{R}^{n}$ with components $\mathbf{f} = (f_1, \dots, f_n)$ is rectifiable if and only if each component is of bounded variation. In this case the proof would break into three parts:

1. Showing the theorem is true for increasing functions.
2. Showing the theorem is true for the difference of two increasing functions (i.e. also for functions of bounded variation)
3. Showing the theorem is true for a function with components of bounded variation.

The third part is fairly simple, I think (just take a partition of $\mathbb{R}^{n}$ that partitions the $k$th dimension as one would partition $\mathbb{R}$ for $f_k$ to obtain a sum of intervals $\lt \epsilon$; the sum of the sizes of the $n$-dimensional intervals covering $\mathbf{f}$ will then be $\lt \epsilon^{n}$), but I'm struggling to make progress with the first two parts. Again, the problem seems to be the possibly countable number of discontinuities.

In any case, I'm not sure how to proceed. Any help will be appreciated.

Answer 1

@Gio67 's answer in the comments:

"If you look on page 33, a path is a continuous function. Apostol defines rectifiability only for paths, so you do have continuity and you can use the answers in your links"

So indeed, the answer to my question can be found here: How prove that the range of an rectifiable curve has measure zero?

N.B. It is on pg. $133$, and not pg. $33$, (at least in my copy of Apostol's "Mathematical Analysis"), that paths are defined.