continuity at a point

[Alex Jordan]

I need to clear up a convention that APEX will use so that I can code the
WeBWorK problems to use that convention.

The book defines "f is continuous at c" as, essentially "f(c) = \lim_{x\to c}f(x),
when both sides exist". However this definition is restricted to c inside an open interval I
on which f is defined.

Does the book consider \lim_{x\to c}f(x) to exist when c is on the edge of f's domain?
There is a separate definition for f being continuous on a closed interval, say [c, d]. But
that shouldn't be used to infer when a function is continuous at c.

The answer to 1.5 #18a is "Yes", so that would say that the intention is for functions to
be considered continuous at endpoints if the value matches the one-sided limit. It's just that
I don't see that spelled out for a student in the section somewhere.

I see two choices (and maybe the group can see more).

1. Define one-sided continuity at a point c. Not my preference purely because it complicates
   the coding of the WeBWorK problems. But if it's the right thing to do, then OK.
2. Define continuity at a point c in f's domain where f is defined on a closed interval [c,d] and
   defined nowhere in some interval (b,c). And also the symmetric situation.
   

[Carly Vollet]

Based solely on the definition of continuity at a point, I would say "No" since f is not defined on an open interval containing 0. However, looking at the source, the answer is marked as "yes", so it seems like Greg is considering some sort of one sided continuity. 

There is a lot of ambiguity/disagreements about what can happen at endpoints. I often times find myself looking at the answer key to determine how an author determines what happens at endpoints. 

Having it explicitly spelled out would be nice. For example, according to the definition of continuity on a closed interval, you would say y=sqrt(4-x^2) is continuous on [-2,2]. But then would you say that it is continuous at x=2 and x=-2? And if "yes", then only because -2 and 2 are the endpoints of the domain. 

Because clearly this doesn't work with something like a step function, y=floor(x) that is continuous on [x,x+1) with x an integer, but not continuous at x. 

I'd like to hear what Greg thinks. I think the easiest answer is to say "no" to continuity at endpoints of the domain, but then explicitly state how a function can be continuous on a [a,b] but not continuous at a or  b. I'd also like to point out that there are previous instances of say, asking  where f is not defined to the left of 0 and the answer key says that this limit is not defined. So saying "no" would be consistent with this definition. 

Carly

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[David Farmer]

An approach that seems natural to me:

1) Define continuity on (a,b) to mean continuous at all c \in (a,b).
(i.e., as you have it now)

2) Define continuity on [a,b] to mean continuous on (a,b) and also
\lim_{x\to b^-} f(x) = f(b) and
\lim_{x\to a^+} f(x) = f(a)

So you don't need to define one-sided continuity.

But there could be an exercise that discusses one-sided continuity and
provides an alternate definition of continuous on an interval.
(There *should* be such an exercise, because having a lot of
exercises is good, but you can start making a wish-list and expect
that someone will come along and contribute it.)

Another exercise: If f is continuous on (a,b) then f is continuous
on [c,d] if a<c<d<b.

Regards,

David

On Sat, 15 Oct 2016, Carly Vollet wrote:

> Based solely on the definition of continuity at a point, I would say "No" since f is not defined on an open interval
> containing 0. However, looking at the source, the answer is marked as "yes", so it seems like Greg is considering some
> sort of one sided continuity. 
> There is a lot of ambiguity/disagreements about what can happen at endpoints. I often times find myself looking at the
> answer key to determine how an author determines what happens at endpoints. 
>
> Having it explicitly spelled out would be nice. For example, according to the definition of continuity on a closed
> interval, you would say y=sqrt(4-x^2) is continuous on [-2,2]. But then would you say that it is continuous at x=2 and
> x=-2? And if "yes", then only because -2 and 2 are the endpoints of the domain. 
>
> Because clearly this doesn't work with something like a step function, y=floor(x) that is continuous on [x,x+1) with x an
> integer, but not continuous at x. 
>
> I'd like to hear what Greg thinks. I think the easiest answer is to say "no" to continuity at endpoints of the domain,
> but then explicitly state how a function can be continuous on a [a,b] but not continuous at a or  b. I'd also like to

> point out that there are previous instances of say, asking \lim_{x\to 0}f(x)  where f is not defined to the left of 0 and

> the answer key says that this limit is not defined. So saying "no" would be consistent with this definition. 
>
> Carly
>
>
> On Sat, Oct 15, 2016 at 11:23 AM, Alex Jordan <jordanc...@gmail.com> wrote:
> I need to clear up a convention that APEX will use so that I can code the
> WeBWorK problems to use that convention.
>
> The book defines "f is continuous at c" as, essentially "f(c) = \lim_{x\to c}f(x),
> when both sides exist". However this definition is restricted to c inside an open interval I
> on which f is defined.
>
> Does the book consider \lim_{x\to c}f(x) to exist when c is on the edge of f's domain?
> There is a separate definition for f being continuous on a closed interval, say [c, d]. But
> that shouldn't be used to infer when a function is continuous at c.
>
> The answer to 1.5 #18a is "Yes", so that would say that the intention is for functions to
> be considered continuous at endpoints if the value matches the one-sided limit. It's just that
> I don't see that spelled out for a student in the section somewhere.
>
> I see two choices (and maybe the group can see more).

> 1. Define one-sided continuity at a point c. Not my preference purely because it complicates

> the coding of the WeBWorK problems. But if it's the right thing to do, then OK.

> 2. Define continuity at a point c in f's domain where f is defined on a closed interval [c,d] and

> defined nowhere in some interval (b,c). And also the symmetric situation.
>
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[Alex Jordan]

On Saturday, October 15, 2016 at 10:35:29 PM UTC-7, David Farmer wrote:

An approach that seems natural to me:

1) Define continuity on (a,b) to mean continuous at all c \in (a,b).
(i.e., as you have it now)

2) Define continuity on [a,b] to mean continuous on (a,b) and also
\lim_{x\to b^-} f(x) = f(b) and
\lim_{x\to a^+} f(x) = f(a)
 

So you don't need to define one-sided continuity.

I think this is exactly how it is right now: Defs 1.5.1 and 1.5.6. But there are
homework exercises that ask is f continuous at the point a, when f is only
defined on [a,... like #14, #18.

Either those exercises should be altered, or there should be some provision
for continuity at a point on the edge of the domain. Maybe in the form of an
exercise/investigation like you suggest.

[Gregory Hartman]

I've been thinking about this a lot today, and it took me a while to appreciate the distinction being asked about. Correct me if I'm wrong: we are trying to differentiate between being "continuous AT a number" vs. "continuous ON an interval." When dealing with open intervals, being continuous ON the interval means it is continuous AT all points in the interval. But what if the interval is closed? Does that imply the function is 'continuous AT the endpoints'? 

When the domain of a function f is [a,b], then it seems natural to me that one could say "f is continuous AT a" when \lim_{x\to a^+} f(x) = f(a). We aren't concerned with what happens when we apply f outside of that closed interval - we shouldn't even try, for as far as f is concerned, nothing else exists but [a,b]. Hopefully this aligns well with an "intuitive" approach to continuity: as x nears a, and eventually x=a, does f(x) behave predictably? This is how I approached the answers to 14, 16 & 18. 

The step function f(x) = floor(x) is given as Example 1.5.4. The answer given is correct using the information known at that point, but in retrospect I wish we had revisited that example after the definition of continuity on closed intervals and updated the answer. I would argue it is more informative to say that f(x) is continuous ON [-2,-1), [-1,0), [0,1), etc., than to say it is just continuous on the related open intervals. By including the left endpoint, we learn information about how f behaves. But we would also say that "f is not continuous AT x=0." 

I think this needs to be spelled out within the text with an example. If done so, the above exercises won't need to be adjusted. 

[Alex Jordan]

Before reading my thoughts on this below, I propose we make an issue on GitHub
and deal with this after Carly and I have completed the manual conversion of the book
(or perhaps of part I). It's looking like addressing the issue involves revising the expository
content, examples, and exercises. At best, it's slows us down to try to work it in now. At worst,
it's a bit confusing and not clear that we would make all the changes in all the right places.

If this sounds good, I'll make the issue on GitHub.

------

I'll document my thoughts here even though I don't believe they are all
relevant to an introductory calculus text. But they do serve to justify a decision
on how to proceed, even though it may seem arbitrary to an intro calc
student/teacher.

If f's domain is D\subset R, then the topological definition of
\lim_{x\to a}f(x)
uses x in open sets of D containing a, minus {a}.

Open sets in D are not necessarily open in R.
Open sets in D are intersections of open sets in R with D.
They can look like closed intervals, half-open intervals,
collections of discrete points, or even weirder things, depending on D.

With this definition, for an f whose domain is [0,1], it's quite possible to say

\lim_{x\to 0}f(x)

exists, without mentioning one-side limits. And it's therefore quite possible
for f to be continuous at 0, even though nothing happens left of 0.

*This argues that the current answers to questions like #14 and #18 are correct.*

For a function like the step function,

\lim_{x\to 0}f(x)

does not exist, and therefore f cannot be continuous at 0. However if someone
asked "is it continuous on [0,1)?", I'd think someone is asking about the function
f restricted to [0,1), with a new domain and a new topology on that domain. And
then the answer is yes.

*This argues that continuity on closed or half-closed intervals is a fine concept that
should be discussed, but points out that it's not enough if you want to talk about
continuity at a point on the boundary of a domain. So some explanation/example(s)
is needed.*

[Gregory Hartman]

I agree we should table this for now and pick it up later. If we have this issue in mind, we can see how it plays out later in the text and hopefully address it in a uniform manner.

Yes, I was thinking of things with a topological mindset (even pulled Munkres out yesterday as I typed my message...). I sense that Alex & I are essentially in agreement, and we'll sort out the finer details later.

[Carly Vollet]

Here is a possible fix. Define continuity differently for points on the interior of the domain vs the ends.

On Oct 18, 2016 5:09 AM, "Gregory Hartman" <gregory...@gmail.com> wrote:

I agree we should table this for now and pick it up later. If we have this issue in mind, we can see how it plays out later in the text and hopefully address it in a uniform manner.

Yes, I was thinking of things with a topological mindset (even pulled Munkres out yesterday as I typed my message...). I sense that Alex & I are essentially in agreement, and we'll sort out the finer details later.

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[Alex Jordan]

OK, issue at
https://github.com/PCCMathSAC/apex-mbx/issues/63

We have several good ideas here to review and apply later.

BTW, yesterday I uploaded new HTML. As far as Carly and I are concerned
(and not counting issues like the one in this thread) we are done with chapter 1
except:

• about half of the 1.6 questions converted to WeBWorK
• I need to make a final decision about what to do with image widths, and
  if it involves hard coding widths into source, then I'll do that through chapter 1
  and make a post here about how to deal with image widths.