Confused about Continuous Endpoints: -1 < a < 1?

In summary: The text says "In order for a function to be continuous at a number a, there must be a continuous function from the interior of the interval to the boundary of the interval." This means that the right-hand limit at the end point does not satisfy this condition.
  • #1
ChiralSuperfields
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Homework Statement
Please see below
Relevant Equations
Please see below
For this problem,
1676490195021.png

I don't understand why they are saying ##-1 < a < 1## since they are trying to find where ##f(x)## is continuous including the endpoints ##f(-1)## and ##f(1)##

Why is it not: ##-1 ≤ a ≤1##

Many thanks!
 
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  • #2
Callumnc1 said:
Homework Statement:: Please see below
Relevant Equations:: Please see below

For this problem,
View attachment 322317
I don't understand why they are saying ##-1 < a < 1## since they are trying to find where ##f(x)## is continuous including the endpoints ##f(-1)## and ##f(1)##

Why is it not: ##-1 ≤ a ≤1##

Many thanks!
Because they forgot about the endpoints.
 
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  • #3
PeroK said:
Because they forgot about the endpoints.
Thank you for your reply @PeroK!
 
  • #4
Callumnc1 said:
Thank you for your reply @PeroK!
They should have done one-sided limits at the end points, in addition to two sided limits at the interior points. As any good maths student will tell you!
 
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  • #5
The part of the proof that you show matches the first line: "If ##-1 \lt x \lt 1##". Is there another part of the proof that you have not shown? If not, then they just made a mistake and left it out.
 
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  • #6
Thank for your replies @PeroK and @FactChecker !

There was a couple of lines that I did not screen shot,
1676494724970.png


However, did they really need those lines if they had just said ##-1 ≤ a ≤1##?

Many thanks!
 
  • #7
Callumnc1 said:
However, did they really need those lines if they had just said ##-1 ≤ a ≤1##?
The theorems and laws that are referenced in the first part are probably stated only for two-sided limits, so it would not be valid to include the endpoints that are one-sided limits. By separating the endpoints, they can use the phrase "similar calculations" and infer similar one-sided theorems and laws.
 
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  • #8
Callumnc1 said:
Thank for your replies @PeroK and @FactChecker !

There was a couple of lines that I did not screen shot,
View attachment 322319

However, did they really need those lines if they had just said ##-1 ≤ a ≤1##?

Many thanks!
So, they didn't forget about the endpoints after all!
 
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  • #9
FactChecker said:
The theorems and laws that are referenced in the first part are probably stated only for two-sided limits, so it would not be valid to include the endpoints that are one-sided limits. By separating the endpoints, they can use the phrase "similar calculations" and infer similar one-sided theorems and laws.
Thank you for your replies @FactChecker and @PeroK!

I think I'm starting to understand. So basically, you can't take the limits of the end points, so you just take the right- and left-hand limits to prove it is continuous.

However, I though you could not do that since the text also states that in order for a function to be continuous at a number a:
1676496403073.png

However, for the end points they only took the right hand or left hand limit for reach end point. How dose that me it is continuous at ## x = -1, 1## (since the limits at each of those end points DNE)?

For example, for ##x = 1## You cannot take the right-hand limit since there is no graph there (so left-hand limit dose not equal right-hand limit, so limit DNE).

I think this could be something to do with Definition 3.
1676496376902.png

Many thanks!
 

Attachments

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1. What does "a < 1" mean in the continuous endpoint?

In this context, "a < 1" indicates that the value of "a" is less than 1. This means that any value between -1 and 1, excluding -1 and 1, can be considered as a possible value for "a".

2. Why is the endpoint -1 excluded in this case?

The endpoint -1 is excluded because the inequality states that "a" must be less than 1. This means that -1 is not a valid value for "a" in this case.

3. What does it mean for an endpoint to be continuous?

A continuous endpoint means that the value can take on any number between two specified values, including fractions and decimals. In this case, "a" can take on any value between -1 and 1, excluding -1 and 1.

4. Can "a" be a negative number in this case?

Yes, "a" can be a negative number as long as it is less than 1. This means that values such as -0.5, -0.75, and -0.999 are all valid values for "a".

5. How does this continuous endpoint affect the interpretation of the data?

The continuous endpoint allows for a wider range of values for "a" to be considered, which can provide a more accurate representation of the data. It also allows for the inclusion of decimal and fractional values, which can be important in certain scientific experiments.

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