Let $f:[a,b] \rightarrow \mathbb{R}^n$ be a rectifiable continuous curve, show that $f[a,b]$ has content zero.
If $f$ is continuous then is integrable so we have that $[a,b]\times f[a,b]$ has content zero for any partition P of $[a,b]$ but how can i show that only The range has measure zero?
As hinted in the comments, you need $n > 1$.
Assuming $n > 1$, to prove that $f[a, b]$ has content zero it suffices to prove that $f[a, b]$ has measure zero, since $[a, b]$ is compact (see here: Measure zero and compact then content zero)
Then a proof that the measure is zero can be found here: $f[a,b]$ has measure zero
