The question states:
Suppose $f$ is founded on $[a,b]$. Suppose also that $f$ is integrable on every closed interval $[c,d]$ contained in the open interval $(a,b)$. Show that $f$ is integrable on $[a,b]$.
So there are two possible courses of action. Either I can attempt to prove that $f$ must be piecewise monotonic on $[a,b]$ by proving that it is monotonic on every open subinterval, or I can attempt to write a proof via the Riemann criterion.
I tried the former idea, but all I know is that $f$ is monotonic on every open subinterval of every [c,d], but I do not know how to bridge the gap between [c,d] and [a,b], despite trying to create a statement of the form $a<c<d<b$ and then finding $a$ and $c$ within $\epsilon$ of each other, and similarly for $b$ and $d$.
Trying to prove it using the Riemann criterion, I struggle to find good functions $s$ and $t$ to use.
The textbook has not introduced limits or continuity yet, nor has it introduced the FTOC. So I must rely solely on basic analytical results involving the Riemann Criterion for the most part.
Finally, I am not looking for a full proof but rather somewhat of an outline/hint as to how I can proceed.
Given $\epsilon>0$, choose $N>0$ large enough such that $(4/N)\cdot(\sup|f|)<\epsilon$ and pick a partition $P=\{a+1/N,...,b-1/N\}$ for $[a+1/N,b-1/N]$ such that $U(f,P)-L(f,P)<\epsilon$, then the partition $Q:=\{a,a+1/N,...,b-1/N,b\}$ for $[a,b]$ is such that \begin{align*} & U(f,Q)-L(f,Q)\\ &=\left(\sup_{[a,a+1/N]}f-\inf_{[a,a+1/N]}f\right)\cdot\dfrac{1}{N}+U(f,P)-L(f,P)+\left(\sup_{[b-1/N,b]}f-\inf_{[b-1/N,b]}f\right)\cdot\dfrac{1}{N}\\ &\leq 4\sup|f|\cdot\dfrac{1}{N}+U(f,P)-L(f,P)\\ &<2\epsilon. \end{align*}