This is exercise 131Xb $(iii)$ from Fremlin Volume 1:
Show that if $f$ is Lebesgue integrable on $\mathbb{R}$, then $(a,b)\mapsto\int_a^b f d\mu$ is continuous. ( Hint: Either consider simple functions $f$ first or consider $\lim_{n\to\infty} \int_{a_n}^b f d\mu$ for monotonic sequences $(a_n)$ ).
Here is my work so far:
Since $f$ is integrable $\int_a^b f d\mu=\int f\chi_{(a,b)} d\mu$ is well-defined in $\mathbb{R}$ for any pair $(a,b)$. Let $a_n \downarrow a$. Then $f\chi_{(a_n,b)}\to f\chi_{(a,b)}$ pointwise. Moreover, $\lvert f\chi_{(a_n,b)}\rvert \leq \lvert f \rvert$ for all $n$, and $\lvert f \rvert$ is integrable because $f$ is. Hence we can apply the Dominated Convergence Theorem to get $\int_a^b f d\mu = \lim_{n\to\infty} \int_{a_n}^b f d\mu$. Since $(a_n)$ is arbitrary, we conclude that $a\mapsto\int_a^b f d\mu$ is continuous from the right. Considering increasing sequences $a_n \uparrow a$ we get $a\mapsto\int_a^b f d\mu$ continuous from the left, and so the map $a\mapsto\int_a^b f d\mu$ is continuous for any $b$. A similar argument shows that $b\mapsto\int_a^b f d\mu$ is continuous for any $a$.
Questions:
$(1)$ How can I show joint continuity with respect to $(a,b)$?
$(2)$ The hint suggests to use monotonic sequences, but why? It seems that considering an arbitrary sequence $a_n\to a$ would do the job, and in fact would cut the proof in half.
Any help is greatly appreciated.