Theorem 7.21 states: If $f:[a,b]\to\mathbb{R}$ is differentiable at every point of $[a,b]$ and $f'\in L^1$ on $[a,b]$, then $f(x)-f(a)=\int_a^x f'(t)dt$ for all $x\in[a,b]$.
From a very early theorem, we get this lower semicontinuous $g$ on $[a,b]$ such that $g>f'$ and $\int_a^b g(t)dt < \int_a^b f'(t)dt + \varepsilon$. Then for any $\eta>0$, Rudin defines the function
$$F_\eta(x)=\int_a^x g(t)dt-f(x)+f(a)+\eta(x-a)\ \ \ \ \ \ \ (a\le x\le b)$$
and later uses the fact that $F_\eta$ is continuous. I know the "$-f(x)+f(a)+\eta(x-a)$" part is continuous. I know if $g$ were continuous, it would be bounded on $[a,b]$, and this would imply the rest of the continuity. But since $g$ is only lower semicontinuous ($\lbrace x:g(x)>\alpha\rbrace$ is open $\forall \alpha\in\mathbb{R}$), I don't see how to get the continuity of $\int_a^x g(t)dt$.