Intervals for increase and decrease

[sean-fitzpatrick]

An item for discussion:

Most calculus textbooks (including ours) have the habit of conflating intervals of increase and decrease with intervals where the derivative is positive or negative. This leads to the use of open intervals when stating solutions to problems asking for intervals of increase and decrease.

But if we think about the definition, it would be more correct to use closed intervals, except for cases where there is a vertical asymptote or some other point missing from the domain.

Possibly this is a case where "mathematically correct" does not imply "pedagogically correct".

In APEX, we correctly point out that if f'(x) is positive on (a,b) and on (b,c), with f'(b)=0, then f(x) is increasing on (a,c).
But if we noted that in fact, f(x) is increasing on (a,b] and [b,c), then the result is immediate.

As an example: for f(x)=x^2, we know that f'(x)>0 on (0,\infty), but in fact f(x) is increasing on [0,\infty), since for any x>0, it is true that f(0)<f(x).

Is it worth making this point in the book?
If yes, is it worth making this point to the extent that we rewrite the exercises so that closed intervals are expected in the answers?

[APEXCalculus]

I've been mulling this over in my head, and here are my most coherent thoughts thus far. In short, I think we should leave things roughly as-is. I think pedagogically there will be confusion over the concepts of: - "what intervals is a function increasing/decreasing",and - "at x=c, is the function increasing or decreasing?". The latter concept isn't defined in the text, but it is a natural thought to consider. The first example of the text, Example 3.3.6, we have critical points at x = -1, 1/3. So we can say "f is increasing on (-\infty, -1 ] and decreasing on [-1, 1/3]". It then appears that at x = -1, f is both increasing and decreasing, though we'd actually explain that at x = -1, f is neither increasing nor decreasing, hence we have a relative max. This conceptual problem is certainly overcomable, but I'm not sure it is worth the extra work with freshmen. I thought our workaround from the last time we discussed similar issues was to specify in the Exercises "Give the *open* intervals on which f is increasing/decreasing." This doesn't deny that f is decreasing on [ -1, 1/3 ] according to our definition of decreasing, but simply we are asking for something else. I'd be open to trying to summarize some of the above as an aside, then checking the text for consistency afterwards. Key Idea 3.3.5 has a bit of an aside within it; perhaps that would become unnecessary with the right aside after Thm 3.3.4.

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On Mon, Oct 27, 2025 at 1:05 PM Sean Fitzpatrick ***@***.***> wrote: *sean-fitzpatrick* created an issue (APEXCalculus/APEXCalculusPTX#385) <#385> An item for discussion: Most calculus textbooks (including ours) have the habit of conflating intervals of increase and decrease with intervals where the derivative is positive or negative. This leads to the use of open intervals when stating solutions to problems asking for intervals of increase and decrease. But if we think about the definition, it would be more correct to use closed intervals, except for cases where there is a vertical asymptote or some other point missing from the domain. Possibly this is a case where "mathematically correct" does not imply "pedagogically correct". In APEX, we correctly point out that if f'(x) is positive on (a,b) and on (b,c), with f'(b)=0, then f(x) is increasing on (a,c). But if we noted that in fact, f(x) is increasing on (a,b] and [b,c), then the result is immediate. As an example: for f(x)=x^2, we know that f'(x)>0 on (0,\infty), but in fact f(x) is increasing on [0,\infty), since for any x>0, it is true that f(0)<f(x). Is it worth making this point in the book? If yes, is it worth making this point to the extent that we rewrite the exercises so that closed intervals are expected in the answers? — Reply to this email directly, view it on GitHub <#385>, or unsubscribe <https://github.com/notifications/unsubscribe-auth/ABT5OFZTRFROAAPDMYGJQKT3ZZGGPAVCNFSM6AAAAACKLDQ75KVHI2DSMVQWIX3LMV43ASLTON2WKOZTGU2TONZWGE2TIOI> . You are receiving this because you are subscribed to this thread.Message ID: ***@***.***>

[sean-fitzpatrick]

I guess in my opinion, it doesn't make sense to talk about a function being increasing or decreasing at a point.

The definition of these terms involves a comparison between two points in an interval. At a single point, it isn't possible to make such a comparison. Of course, a derivative can be positive or negative at a point, and this is the source of confusion: students forget that there was a definition, and conflate "increasing" with "f'(x)>0".

If you ask students to define the word "increasing" on a test, they will answer with something about the derivative. But you can define what it means for a function to be increasing if it isn't even continuous, let alone differentiable.

In any case, I did some checking, and it looks like most of the exercises are programmed to accept both open and closed intervals for the answers.