LT4 - continuity

[AbbyANoble]

(copying this from slack so you have it documented!)

I've found (or rather, a student has found!) an issue with LT4 in CheckIt. On the ones, like #14, where there is supposed to be a removable discontinuity, the function is actually continuous at that point. There is no hole because one side uses <= (or >=).

A fix -- on the removable ones make sure the "or equal to" does not come up. And then on all the others, maybe occasionally leave it off so students don't associate removable discontinuities as the only spot where they would say neither. Also update the instructions to say "The function h(x) is discontinuous at x=-3. Determine if h(x) h(x) is left- or right-continuous (or perhaps neither) at x=-3 and then classify the type of discontinuity present at x=-3."

[StevenClontz]

Not a huge fan of "left/right-continuous" as a concept. I also note that you don't define those ideas in https://stevenclontz.github.io/cal1-tbil-2021/LT4.html so should I just remove that part?

Would it make sense to always have part (b) have three cases, x<a, x=a, and x>a? Then we can make sure either x<a and x=a match, x=a and x>a match, or x<a and x>a match, with the third always not matching, so it's not continuous. Then it's a jump discontinuity in the first two cases, and removable in the last.

[StevenClontz]

Here's an implementation:

[StevenClontz]

(adding back left/right continuity isn't too hard I guess)

[AbbyANoble]

I can go either way on left or right continuous. I just went with it and talked about it when we did a Check It problem in class.

I like the solution having the <, =, > in there. Makes it less obvious, and I don't think it's too much of a shift for my students since they have already started these.

Are the infinite discontinuities still getting generated? I wouldn't want to lose those.

[StevenClontz]

Infinite discontinuity was lost (assuming it was there originally; I ended up rewriting). Remark 1.4.12 https://stevenclontz.github.io/cal1-tbil-2021/LT4.html#remark-3 didn't mention it so I didn't take care to make it a possibility. I'm not sure exactly how to generate that case without the issue of having the pattern of only showing fractions for infinite discontinuities.

[AbbyANoble]

Linking this in slack for other calc 1 people to weigh in!

Yeah. I think there is some disconnect through a lot of our writing and the CheckIt problems, not unique to this section. I don't know that I realized the infinite wasn't in the activity book, but they did do some in class when they worked on CheckIt problems.

In the other version, jump discontinuities and infinite both had one side as a fraction. See below. Just on the infinite the x-value was an asymptote and the other was not.

[fragandi]

TO-DO items for LT4 seem to be

1. Rewording prompt a. to elicit explanation
2. Rewording prompt b. to elicit why the function is not cont in the sense of what condition for continuity fails and then classify the type of discontinuity
3. (already done?) Eliminate right and left-continuous language. I think that we get to the same concept with the rewording from item 2.

[fragandi]

I have done the items above. @AbbyANoble check and close the issue if ok!