How can one show that this integral is Riemann integrable?

Question

It is given that $$ f:[a,b] \mapsto C~ \text{is Riemann integrable with b>a}$$ and i have to show that $$ \overline{f}:[a,b] \mapsto C~ \text{is Riemann integrable} $$ and $$\displaystyle \int_{a}^{b} \overline{f(x)} \,dx=\overline{\displaystyle \int_{a}^{b}f(x)\,dx }$$

Answer 1

The Riemann sum, $$\int_a^b f(x)dx=\lim_{n\to \infty}\sum_{k=1}^n f(\xi_k)(x_{k}-x_{k-1})\;\;\mbox{(the limit is taken in } \mathbb C),$$ where $a=x_0<x_1<\cdots<x_n=b$ and $\xi_k \in [x_k, x_{k-1}]$ with $\max_{1\leqslant k \leqslant n}|x_{k}-x_{k-1}|\to 0$ as $n\to \infty$. Now, by the properties of conjunction $$\lim_{n\to \infty}\sum_{k=1}^n \overline{f(\xi_k)}(x_{k}-x_{k-1})=\overline{\lim_{n\to \infty}\sum_{k=1}^n f(\xi_k)(x_{k}-x_{k-1})}=\overline{\int_a^b f(x)dx}.$$ These equalities imply that $\overline{f}$ is also Riemann integrable and $\int_a^b \overline{f(x)}dx = \overline{\int_a^b f(x)dx}$.