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

• ChiralSuperfields
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.
ChiralSuperfields
Homework Statement
Relevant Equations
For this problem,

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!

Callumnc1 said:

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.

Mark44 and ChiralSuperfields
PeroK said:
Because they forgot about the endpoints.

Callumnc1 said:
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!

ChiralSuperfields
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.

ChiralSuperfields
Thank for your replies @PeroK and @FactChecker !

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

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

Many thanks!

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.

PeroK and ChiralSuperfields
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!

ChiralSuperfields
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:

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.

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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