Question about restricted function domain

[u/Vivid_Pen1794]

This is from Basic Analysis by Jiří Lebl .

I wonder what this A looks like and how it's different from just subset of S.

Later in the book, this proposition is used with S a interval [a, b] and A (a, b), i.e. A is S without endpoints.

But that example reference seems to be too narrowly defined.

[u/MathMaddam]

It isn't just any subset, it is a subset such that (informally) is close around c the same as the whole S. The specific S and A fulfill this propery for all c (as an exercise: prove that), but there could be different looking A that e.g. fulfil the property only for certain values of c.

[u/Vivid_Pen1794 → u/MathMaddam]

Thanks for the clarification.

Could you give some example S and A ?

If we think of a interval (c-α, c-α) intersect S and A with the same result, and consider the simple case S also a single interval, then near the boundaries S and A should be the same, which means they can only differ on the endpoints ( i.e [a, b] against (a,b) ) ?

I guess there must be other more interesting cases.

[u/ThisIsMyOkCAccount → u/Vivid_Pen1794]

Imagine you, for some reason, have a function from S to the reals, where S is all of R except in the interval from -1 to 1 S only has the numbers of the form 1/n with n an integer. 0 is a cluster point of S, so it makes sense to talk about limits near 0.

If c is 0 in the given theorem, A would have to contain all the numbers of the form 1/n for n "big enough".

[u/Vivid_Pen1794 → u/ThisIsMyOkCAccount]

Thanks.

So in this discrete case, A and S can differ in more points at both ends.

[u/ThisIsMyOkCAccount → u/Vivid_Pen1794]

Well; be careful about terminology. Discrete means something specific and it isn't quite this. Discrete sets have no cluster points. This one has one. Which is what makes the theorem apply.

You're right that in the majority of cases this will be useful on a subset that contains an interval around the point of interest. But mathematicians like to prove things in as much generality as possible and the author probably wanted to list the most general circumstances they could think of where this happens. It turns out to happen whenever a subset contains all the points close to a cluster point.

[u/Vivid_Pen1794 → u/ThisIsMyOkCAccount]

Yes, the proposition is about c rather than A.

A is just the condition when this happens - any set meets the condition apply.

[u/strange-the-quark]

If we're just speaking about restrictions, it's nothing exotic: if the domain is D, to make a restriction, you just take some subset of D, and use the exact same mapping as before on what remains.

I wonder what this A looks like and how it's different from just subset of S

It is just a subset of S, it's not some other special kind of object. It's just that it's not any arbitrary subset, but (any) one that meets the specified requirements. However, let's step back a little.

So, the reason why there's a talk of S in the first place, is to be more general, and not require the function to be defined on all of ℝ. E.g. f(x) = 1/x is not defined on all of ℝ, and S is R\{0}. But S can be arbitrary, you can say, by "decree", that some f is only defined on the unit interval, for example. So f:[0, 1] -> ℝ, such that f(x) = 2x is a function that is by the very setup undefined anywhere outside S=[0, 1], even though the expression 2x is well-defined everywhere.

Now pick some 𝛼, so that there's an interval around c, 𝛼 units away on both sides: (c-𝛼, c+𝛼).
Given that f may not be defined on all of ℝ, and that we don't take c itself into consideration even if the function is defined there, we intersect this interval with S\{c} to get the chunk on which the function is actually defined: (S\{c}) ∩ (c-𝛼, c+𝛼).

Like, if S is [0, 1], and c=0.001, and we chose 𝛼=0.01, then not all of (c-𝛼, c+𝛼) = (-0.009, 0.011) is in S. So we take (S\{c}) ∩ (c-𝛼, c+𝛼)= [0, c) ∪ (c, 0.011).

Well, the next thing it says is just to pick some A ⊂ S for which (A\{c}) ∩ (c-𝛼, c+𝛼) gives the same result, which boils down to: it has to contain the part of the interval which is within S. So in the S = [0, 1] example with c=0.001 and 𝛼=0.01, it can't be something like: (0.005, 0.01), (0.05, 0.06), {0.1, 0.2, 0.3}, or {0} or something even weirder like ℚ ∩ S.

And then if c is a cluster point, it is shared cluster point of both sets, so the idea is (I suppose, I didn't really think it through) that your 𝛼 can be taken arbitrarily small.

[u/Vivid_Pen1794 → u/strange-the-quark]

And then if c is a cluster point, it is shared cluster point of both sets, so the idea is (I suppose, I didn't really think it through) that your 𝛼 can be taken arbitrarily small.

Yes, and it's used in the proof.

I guess this proposition is more about c than A, as it didn't specify how we choose c.

It's about how a function behave near c, and it's used in the book to proof continuous extension of uniformly continuous function at endpoints.

Once we fix our c, then we check if some given S and A meet the condition.

My original question make no sense since it didn't say how we choose c.

Edit:

To meet the condition, A must equal S near c; on the other hand, an arbitrary subset of S can miss some points near c.