# a theorem to simplify continuity in Stewart's calculus: early transcendentals

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:

$$1.~~~~~~~ f(x)+g(x)$$ $$2.~~~~~~ f(x)-g(x)$$ $$3.~~~~~~ f(x)g(x)$$ $$4.~~~~~~ c\cdot f(x)$$

$$5. ~~~~~~ f(x)/g(x)$$

$$6.~~~~~~f(g(x)).$$

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.  • I think the only difference is passing off where the discussion of a restricted domain occurs. Do you want to have that discussion in the context of each example, like Examples 6 and 7 that you shared? Or do you want to have the discussion when introducing the theorem you proposed and (perhaps) attempting to explain a proof of that theorem? If you just want students to "efficiently apply a theorem" from what it says, then surely what you suggest is "faster". But is it better, in the sense of student understanding? I'm not so sure. – Brendan W. Sullivan Sep 13 '18 at 20:34
• @BrendanW.Sullivan thanks for the feedback. I'm not so sure either. I think I may do them both ways. I just wanted to be sure I'm not setting some kind of trap for myself down the line. – Tim kinsella Sep 13 '18 at 20:39
• No trap, it's all logically equivalent. It's just a matter of whether you want to "do the work" in examples or proofs. – Brendan W. Sullivan Sep 13 '18 at 20:41
• @BrendanW.Sullivan Could you write that as an answer? – Tommi Brander Sep 14 '18 at 6:05
• I just read this section and there's nothing complicated in it. In (1), the author defines local continuity using limits, in (2) he defines left and right local continuity, in (3) he defines global continuity and (4) is the theorem you wrote about sums, products, etc. of continuous functions. – Paracosmiste Sep 14 '18 at 15:34