Proof of the linearity of complex integrals for paths of bounded variation? I am familiar with the proof of the linearity of complex integrals for piece-wise smooth paths. Nonetheless, complex integrals can be defined for more general paths $\gamma:[a,b]\to\mathbb{C}$ where $\gamma$ is of bounded variation (and not necessarily (piecewise)smooth). Let $f$ and $f$ be continuous functions from $[a,b]$ to the complex plane. which is the proof of
$$
\int_a^b\alpha{}f(z)\,dz=\alpha{}\int_a^bf(z)\,dz\qquad\int_a^b\bigg(f(z)+g(z)\bigg)\,dz=\int_a^bf(z)\,dz+\int_a^bg(z)\,dz
$$
 in this more general setting?
 A: To whom it may concern. Let us name
$$
\int_a^bf(z)\,dz\equiv{}I_1\qquad\int_a^b\alpha{}f(z)\,dz\equiv{}I_2
$$
Let's pick any $\epsilon_1>0$. By definition there is a $\delta_1$ such that for any partition $P={a=t_0<t_1<\ldots<t_m=b}$ satistying $||P||<\delta_1$ (where $t_{k-1}\leq\tau_k\leq{}t_k$)
$$
\bigg|\sum_{k=1}^mf(\tau_k)\big[\gamma(t_k)-\gamma(t_{k-1})\big]-I_1\bigg|<\epsilon_1
$$
by definition also for an $\epsilon_2<0$ there is a $\delta_2<0$ such that for a partition satisfying $||P||<\delta_2$
$$
\bigg|\sum_{k=1}^m\alpha{}f(\tau_k)\big[\gamma(t_k)-\gamma(t_{k-1})\big]-I_2\bigg|<\epsilon_2
$$
Let's make a guess and check if the definition is satisfyed by $I_2=\alpha{}I_1$.
$$
\bigg|\sum_{k=1}^m\alpha{}f(\tau_k)\big[\gamma(t_k)-\gamma(t_{k-1})\big]-\alpha{}I_1\bigg|=\bigg|\sum_{k=1}^mf(\tau_k)\big[\gamma(t_k)-\gamma(t_{k-1})\big]-I_1\bigg|\big|\alpha\big|<\big|\alpha\big|\epsilon_1
$$
where to write the last inequality we assumed that $||P||<\delta_1$. Therefore we see that for every $\epsilon_2=|\alpha|\epsilon_1$ there is a $\delta_2=\delta_1$ such that the definition is satisfied for $\alpha{}I_1$ being the value of the integral.
Now the sum. same as before by definition 
$$
\bigg|\sum_{k=1}^mf(\tau_k)\big[\gamma(t_k)-\gamma(t_{k-1})\big]-I_1\bigg|<\epsilon_1
$$
$$
\bigg|\sum_{k=1}^mg(\tau_k)\big[\gamma(t_k)-\gamma(t_{k-1})\big]-I_2\bigg|<\epsilon_2
$$
Notice that the epsilons an deltas have nothing to do with the ones  from the first part. Let's now consider
$$
\bigg|\sum_{k=1}^mf(\tau_k)\big[\gamma(t_k)-\gamma(t_{k-1})\big]+g(\tau_k)\big[\gamma(t_k)-\gamma(t_{k-1})\big]-I_1-I_2\bigg|\leq{}\bigg|\sum_{k=1}^mf(\tau_k)\big[\gamma(t_k)-\gamma(t_{k-1})\big]-I_1\bigg|+\bigg|\sum_{k=1}^mg(\tau_k)\big[\gamma(t_k)-\gamma(t_{k-1})\big]-I_2\bigg|<\epsilon_1+\epsilon_2
$$
wehre i have considered $||P||<inf(\delta_1,\delta_2)$. Therefore for $\epsilon=\epsilon_1+\epsilon_2$ and $\delta=inf(\delta_1,\delta_2)$ the definition is satisfied. Thus, the complec integral is linear.
