Let $-∞<a<b<+∞$ and $k∈ℕ$. If a function $γ:[a,b]→ℝ^k$ has bounded derivative on $[a,b]$,
(a) prove that $\gamma$ is rectifiable.
(b) If $\gamma'$ is Riemann integrable on $[a,b]$, determine whether the equality $Λ(γ)=\int_a^b|γ'(t)|dt$ is valid or not.
I have already proved the problem (a), just using a theorem "$∃c∈(a,b), |f(b)-f(a)| ≤ (b-a)|f'(c)|$" where $f:[a,b]→ℝ^k$ is continuous on $[a,b]$ and differentiable on $(a,b)$. And I also know if $γ'$ is cont on $[a,b]$, then $Λ(γ) = \int_a^b|γ'(t)|dt$. But, what about the case that $γ'∈$ on $[a,b]$? Are two values $Λ(γ)$ and $\int_a^b|γ'(t)|dt$ same each other?
Reference: Principles of Mathematical Analysis - 3rd Edition by Walter Rudin, 137p