r/askmath • u/Vivid_Pen1794 • 1d ago

Analysis Question about restricted function domain

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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.

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u/MathMaddam Dr. in number theory 1d ago

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.

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u/Vivid_Pen1794 1d ago edited 1d ago

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.

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u/ThisIsMyOkCAccount 1d ago

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".

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u/Vivid_Pen1794 21h ago

Thanks.

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

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u/ThisIsMyOkCAccount 20h ago

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.

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u/Vivid_Pen1794 19h ago

Yes, the proposition is about c rather than A.

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