This is the definition from my book and I don't quite understand what does ' all closed intervals $[c,d]$ ' mean. Does that mean there can be an arbitrary number of such intervals, all which do not overlap each other, and for which there is an assigned regulated function to all of these intervals? I think I'm confused because later in the definition it writes that $c$ and $d$ tend to the right limit of $a$ and left limit of $b$.(Hence there can only be one interval $[c,d]$ ?)
Also, what would happen if we did not take c to the right limit of a and d to the left limit of b. As in, why are we taking the limits of these intervals to be $a+$ and $b-$ ? Can't we just integrate it from a to b?
No, you cannot simply integrate from $a$ to $b$ since $\int_a^bf(x)\,\mathrm dx$ is undefined (unless $[a,b]$ is a subset of the domain $D_f$ of $f$). On the other hand, if $[c,d]\subset D_f$, then $\int_c^df(x)\,\mathrm dx$ is defined. And so basically we defined $\int_a^bf(x)\,\mathrm dx$ as the limit of these integrals (when $c$ goes to $a$ and $d$ goes to $b$).