Following the Brézis book in French we have the following proof that $L^p(\Omega)$ is separable for $1 \leq p <\infty$.
I have a doubt in the construction of the function $f_2$. Following the proof, I tried to complete the details as follows:
Since $f_1$ is continuous, then the oscillation of $f_1$, $\omega(f_1,x)=\lim_{\delta \rightarrow 0 }\omega(f_1,\Omega \cap B(x,\delta))$ in each point $x \in \Omega'$ is zero. Therefore, for each $x \in \Omega'$ there exists $R_x=\prod_{k=1}^{N}]a_k,b_k[ \subset B(x,\delta)$, $R_x \ni x$ such that $|\omega(f_1,x)|<\frac{\varepsilon}{2|\Omega'|^{1/p}}$ for all $x \in R_x$. Thus, $\cup_{x \in \Omega'}R_x$ is a open covering of $\hbox{supp} f_1$, which is compact. Therefore, there exists $n_0 \in \mathbb{N}$ and $x_1,\dots,x_{n_0}$ such that $\hbox{supp} f_1 \subset \cup_{i=1}^{n_0}R_{x_i}$. Following this question, consider $C_i \in[\inf_{R_{x_i}}f_1,\sup_{R_{x_i}}f_1]$. Thus, $$|C_i\chi_{R_{x_i}}(x)-f_1(x)|<\frac{\varepsilon}{2|\Omega'|^{1/p}}$$ for all $x \in R_{x_i}$.
My question: Let $f_2(x)=\sum_{i=1}^{n_0}C_i\chi_{R_{x_i}}(x)$ as in the question linked, can we conclude that $\|f_1-f_2\|_{\infty}<\varepsilon$?
The biggest problem I see is that cubes can have non-empty intersection.