I'm reading page 4 of Fourier Series and Boundary Value Problems 8th edition by James Ward Brown and Ruel V. Churchill -
"If two functions $f_1$ and $f_2$ are each piecewise continuous on an interval $a<x<b$, then $\color{red}{\text{there is a finite subdivision of the interval}}$ such that both functions are continuous on each closed subinterval when the functions are given their limiting values from the interior at the endpoints. Hence linear combinations $c_{1}f_1 + c_{2}f_2$ and products $f_1f_2$ have that continuity on $\color{orange}{\text{each subinterval}}$ and are themselves piecewise continuous on the interval $a < x < b$."
It seems like the authors are saying that there is a finite set of closed subintervals in $a < x < b$ that these two functions are both continuous on, but not necessarily all possible subintervals, which allows for their sum and product to both be continuous on the entire interval.
Then in the next sentence, with the "each subinterval" highlighted in orange, they are saying that the product and sum of the two functions hence have continuity on each of the subintervals of $a< x < b$.
So what I'm getting form this is that the functions themselves have to be continuous on only some of the subintervals, and that this allows their product and sum to be continuous on all of the subintervals?