$\quad$ Let a finite closed interval be $[a,b]$. Consider the absolutely continuous function space $\mathrm{AC}([a,b])$ on it, with a norm as follows:
$$\|f\|_{\mathrm{AC}}=\sup_{x\in[a,b]}|f(x)|+\|f'\|_{\mathcal L^1([a,b])}$$
$\quad$ How to prove that $\mathrm{AC}([a,b],\|\cdot\|_{\mathrm{AC}})$ is separable?
$\quad$ I already know that this space is Banach, and I want to find a countable dense subset of it. I've tried using Weierstrass approximation theorem, and found polynomials whose coefficients are rational, but I can't say that $\|f'-p'\|_{\mathcal L^1([a,b])}<\varepsilon$. Although I know that this is equal to the total variation of a function, it doesn't help a lot. Is there a problem with my direction? If yes, how do you find such a subset?
Since continuous functions are dense in $L^{1}$ Weierstrass Theorem shows that there is a polynomial $q$ such that $\|f'-q\|_{L^{1}}<\epsilon$. Let $p(x)=f(a)+\int_a^{x}q(t)dt$. Then $p$ is a polynomial, $p'=q$ and $\|f'-p'\|_{L^{1}}<\epsilon$. Also, $|f(x)-p(x)|=|[f(a)+\int_a^{x}f'(t)dt]-[f(a)+\int_a^{x}q(t)dt]|\le \int_a^{b}|f'-q|<\epsilon$ for every $x$. Now approxinate $p$ by a polynomial with rational coefficients.