In show that a straight line has a Lebesgue measure of zero, does the proof given applies to every $C^1$-function or does it need some changes for it to be true?
More specifically, I want to know if the argument that $λ(K)≤∑_i(b_i−a_i)\frac{2ϵ}{2^i}$ works for $f(x)$ as $C^1$-function?
You are correct. I think the following works:
It is a general fact that if $f:\mathbb R^n \to \mathbb R$ is a $C^1$ function, then the graph $$\Gamma(f):=\{(x,v) \mid f(x)=v\}$$ is a $C^1$ submanifold of codimension $1$ in $\mathbb R^{n+1}$. Indedd, the map $f:\mathbb R^n \to \Gamma(f)$, $x \mapsto (x,f(x))$ provides a $C^1$-diffeomorphism to the graph, so it is enough to see that hyperplanes have measure zero, so the last proof applies.