# Show that ##1 - \sqrt {1 - x^2}## is continuous on the interval ##[-1, 1]##

Callumnc1
Homework Statement:
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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!

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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 Callumnc1
Callumnc1
Because they forgot about the endpoints.

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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!

Callumnc1
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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.

Callumnc1
Callumnc1
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!

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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 Callumnc1
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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!

Callumnc1
Callumnc1
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!

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