calculus - What is the significance of proving the integrability of monotonic functions?

The complete question is

What is the significance of proving the integrability of monotonic functions when it was proved the integrability of a general one?

Doesn't the integrability of a general function implies that of a monotonic one?

I'm reading Calculus I, which says:

Theorem 1.12: If $g$ is monotonic on a closed interval [a,b], then $g$ is integrable on $[a,b]$.

But by Theorem 1.9 we stablished the existence of the integral of a general function $f$ bounded on a closed interval iff $\underline{\mathbf{I}}(f) = \bar{\mathbf{I}}(f)$.

I don't think the proof of Theorem 1.12 seems to care about the character of $g$ as a monotonic function. It looks like one can stuck that proof with Theorem 1.19 and it would work equally well.

Certainly a general function $f$ whose domain is $[a,b]$ cannot be said to be monotonic, but, I think, one always can choose an arbitrary partition $P$ of $[a,b]$ with $n$ subintervals $[x_{k-1}, x_k]$ where $f$ is either increasing or decreasing and therefore monotonic on $[x_{k-1}, x_k]$.

In the way I'm (mis)understanding this, I would show the integrability of a general function $f$ on a closed interval (Theorem 1.19 does that), and then show that a set of monotonic functions is a subset of a set of general ones, which implies they are integrable too.

Well, I haven't dedicaded further thoughts to the last proposal, but I haven't been able to see the difference between the two cases either.