I have some confusions in the proof of the following theorem.
If $\gamma:[a,b]\to\mathbb C$, for $[a,b]\subset\mathbb R$, is piecewise smooth then $\gamma$ is of bounded variation and its total variation is given by $$V(\gamma)=\int_{a}^{b}|\gamma'(t)|\,dt.$$
In the proof, the author argues as follows
My Questions:
- In the second equality, he uses the Fundamental Theorem to write $\gamma(t_k)-\gamma(t_{k-1})=\int_{t_{k-1}}^{t_k}\gamma'(t)\,dt$. But why is this true for complex-valued functions $\gamma$ ? The author didn't prove any analog of the FTC for complex functions (yet). Does this follow for some other reasons?
- In the second paragraph, why does $\gamma'$ being continuous imply that $\gamma'$ is uniformly continuous? I thought this was, in general, only true for the converse.
The rest of the proof makes sense to me except these two arguments. TIA.
The FTC is true for complex, and more generally $\mathbb{R}^n$ valued functions because you can apply the usual FTC to each component function.
$\gamma'$ is a continuous function on a compact set. Uniform continuity is a consequence. See proposition 3.1.4 on page 90 of http://mtaylor.web.unc.edu/wp-content/uploads/sites/16915/2018/04/anal1v.pdf