I'm covering section 2.5 of Stewart (on continuity) and stewarts treatment seems needlessly complicated. It seems like the following theorem would streamline a lot of it:
If $f(x)$ and $g(x)$ are functions which are continuous at every point in their domains, then each of the following functions is continuous at every point in its domain:
$$1.~~~~~~~ f(x)+g(x)$$ $$2.~~~~~~ f(x)-g(x)$$ $$3.~~~~~~ f(x)g(x)$$ $$4.~~~~~~ c\cdot f(x)$$
$$5. ~~~~~~ f(x)/g(x)$$
Is there something pedagogically wrong with giving this theorem?
It seems like it would simplify examples like the ones below because it reduces them to just finding the domain of the function.