# Limits of functions and sequences

• I
• trees and plants
In summary, the conversation discusses the concept of functions and limits in different mathematical spaces, including euclidean spaces and topological spaces. The topic of continuity and differentiation is brought up, as well as the possibility of a function having limits at points. The existence and uniqueness of limits in different types of spaces, such as metric spaces and Hausdorff spaces, is also discussed. The conversation ends with a mention of the Zariski topology and its implications on the concept of limits in algebraic geometry.
trees and plants
Hello there.Is there any function or sequence that has no limits at any point? I am not necessarily talking about functions on euclidean spaces, they could be on topological spaces in general.Also, we have homeomorphism that is about I think mostly continuity, diffeomorphism about differentiation could we define a function that allows for a function to have limits at points? Thank you.

Let f(x)=1 if x is rational number, 0 if x is irrational number.
For any real number x, f(x) is 0 or 1 but lim ##x\rightarrow a## f(x) does not exist for any real number a.

Delta2 and WWGD
Thank you.So, we could define a function that allows for having limits in functions of a mathematical space?But I thought of something, topological spaces have open sets, so they allow limits on functions?Is this incorrect?

Ordinary functions, e.g. f(x)=1,f(x)=sin x, are continuous and have limits at any point. Maybe I do not catch you correctly.

And what about functions that are not continuous?Could they not have limits on any point?Like the one you said before?

If there is a discontinuity, no limit exists on that point.

anuttarasammyak said:
If there is a discontinuity, no limit exists on that point.

That's true of the specific function in post #2. However, in general, if ##f(x)## is discontinuous at ##x = a## then it is possible for ##\lim_{x \rightarrow a} f(x) = L ## to exist. The fact ##f(x)## is discontinuous at ##x = a## only says that ##L \ne f(a)##.

Delta2, WWGD and anuttarasammyak
For an example
f(x)=1 for x ##\neq## 0, 0 for x=0
f(0)=0
lim ##x \rightarrow 0## f(x)=1

As I know a limit is more general than a function being continuous at a point. If there are more functions that have discontinuity and have limits but others that have discontinuity but do not have limits then perhaps a function that allows for a function to have limits at points could be useful and have applications. What do you say? Something else I wanted to say is that on topological spaces some sequences may have more than one limit.Are there theorems about the number of those limits?Or when there is only one limit of the sequence? Thank you.

For your possible interest say function f(x) have finite points of discontinuities at ##x=a_i##, ##i=1,2,...## as
$$\lim_{x\rightarrow a_i-0}f(x)=b_{i-}$$
$$\lim_{x\rightarrow a_i+0}f(x)=b_{i+}$$
$$f(a_i) =b_i$$
Except the points of discontinuity the function is expressed by Fourier integrals whose value at the discontinuities are
$$f(a_i)=\frac{b_{i+} +b_{i-}}{2}$$
As an example of function in post #9, Fourier integral function is f(x)=1 for any x.

universe function said:
Something else I wanted to say is that on topological spaces some sequences may have more than one limit.
It's been decades since I studied anything to do with topological spaces, but I don't recall anything about sequences having more than one limit. It's possible for a sequence to have subsequences that each have their own limit. For example, ##s_n = \{(-1)^n\}, n = 1, ..., \infty## is a divergent sequence, but one subsequence converges to 1, and the other converges to -1.

Mark44 said:
It's been decades since I studied anything to do with topological spaces, but I don't recall anything about sequences having more than one limit. It's possible for a sequence to have subsequences that each have their own limit. For example, ##s_n = \{(-1)^n\}, n = 1, ..., \infty## is a divergent sequence, but one subsequence converges to 1, and the other converges to -1.
In the trivial topology of the universal set itself and the empty set all sequences converge to all points.

S.G. Janssens
In general topological spaces, it is often more useful to work with nets, also known as Moore–Smith sequences, although they are not sequences, but rather generalizations thereof.

I say it is "more useful", because the well-known characterizations of e.g. continuity and compactness in terms of sequences break down in general topological spaces, but they do hold in terms of nets.

Regarding the question about sequences having more than one limit, the extreme example is indeed as in post #13. There is a theorem that says that every net has at most one limit if and only if the topological space is Hausdorff. (Intuitively, being Hausdorff means that the space has enough open subsets.)

You can find elaborations of this in most books (with a good chapter) on point-set topology. If specific references are needed, let me know.

PeroK
Mark44 said:
It's been decades since I studied anything to do with topological spaces, but I don't recall anything about sequences having more than one limit.
No, but a sequence can have one or more cluster points. A sequence has a limit iff it has one and only one cluster point.

Svein said:
A sequence has a limit iff it has one and only one cluster point.
In general topological spaces this need not be true. Post #13 is a counterexample for this, as well.

More specifically,
##\Longrightarrow## need not be true in general topological spaces.
##\Longleftarrow## need not be true even in metric spaces.

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You have made me realize another odd fact about the zariski topology that we use in algebraic geometry. On any affine space, say the real line, zariski closed sets are zero loci of polynomials, hence either the whole line, or finite sets. Hence every sequence consisting of distinct points, converges in this topology to every point of the line. Needless to say we don't use sequences much in this topology.

S.G. Janssens
mathwonk said:
You have made me realize another odd fact about the zariski topology that we use in algebraic geometry. On any affine space, say the real line, zariski closed sets are zero loci of polynomials, hence either the whole line, or finite sets. Hence every sequence consisting of distinct points, converges in this topology to every point of the line. Needless to say we don't use sequences much in this topology.
Yes, it has been my understanding that Zariski topology is one of the motivations in some textbooks to not restrict attention to Hausdorff spaces too early in the course.

universe function said:
As I know a limit is more general than a function being continuous at a point. If there are more functions that have discontinuity and have limits but others that have discontinuity but do not have limits then perhaps a function that allows for a function to have limits at points could be useful and have applications. What do you say? Something else I wanted to say is that on topological spaces some sequences may have more than one limit.Are there theorems about the number of those limits?Or when there is only one limit of the sequence? Thank you.
In a Hausdorff topological space, sequences or nets may converge to only one limit . Otherwise, you can have as many limit points as you wish ( countably-infinite, I believe), by unioning sequences with different limits. The LimSup, LimInf of a sequence will output the largest and smallest limit points of the sequence respectively. If the two are equal to , say, L, then the sequence converges to L.

Indeed in scheme theoretic algebraic geometry the Zariski topology on a scheme is not even T1, much less T2 (Hausdorff). I.e. points need not be closed, and on an irreducible scheme say, there is even a "dense" point, whose closure is the whole scheme, and which thus lies in every non empty open set. In the more classical setting of algebraic varieties, points are closed, i.e. classical varieties are T1, but Hausdorffness still fails as in the example in post #17 above.

mathwonk said:
You have made me realize another odd fact about the zariski topology that we use in algebraic geometry. On any affine space, say the real line, zariski closed sets are zero loci of polynomials, hence either the whole line, or finite sets. Hence every sequence consisting of distinct points, converges in this topology to every point of the line. Needless to say we don't use sequences much in this topology.
I don't see why sequences converge to every point on the line?

WWGD said:
In a Hausdorff topological space, sequences or nets may converge to only one limit . Otherwise, you can have as many limit points as you wish ( countably-infinite, I believe), by unioning sequences with different limits. The LimSup, LimInf of a sequence will output the largest and smallest limit points of the sequence respectively. If the two are equal to , say, L, then the sequence converges to L.

I think you can get uncountably many limit points. For example let your sequence be 1/2, 1/4,2/4,3/4,1/8,2/8,...

Then every real number in [0,1] is a limit point of this sequence.

Office_Shredder said:
I think you can get uncountably many limit points. For example let your sequence be 1/2, 1/4,2/4,3/4,1/8,2/8,...

Then every real number in [0,1] is a limit point of this sequence.
Then by Bolzano-Weirstrass ( Every bounded , infinite subset of Euclidean space has a limit point), the set of limit points itself would have a limit point.

Office_Shredder said:
Then every real number in [0,1] is a limit point of this sequence.
A cluster point.

WWGD said:
I don't see why sequences converge to every point on the line?
It was about sequences of distinct points. (Then it is probably clear: Take any point ##x## in the space. An open nbh of ##x## is the complement of a finite set, and a sequence of distinct points leaves such a finite set eventually.)

WWGD
Svein said:
A cluster point.
https://en.m.wikipedia.org/wiki/Limit_point

a limit point (or cluster point

I think those are the same thing? Confusingly, a limit point is not the same as the limit of a sequence.

Edit: ah, perhaps I am wrong. That's the definition of a limit point of a set, and then it goes on to talk about cluster points for sequences without re-using that word.

I really don't see how that contradicts the claim that in at least some contexts the word "limit point" means the same thinghttps://www.emathzone.com/tutorials/real-analysis/limit-points-of-a-sequence.html#:~:text=A number l is said,values of n∈N.
As in the case of sets of real numbers, limit points of a sequence may also be called accumulation, cluster or condensation points.

So there's at least one more source that agrees with my hazy recollection. "Limit point" does not mean "Limit of the sequence", and actually means "cluster point".

At any rate, this is a matter of semantics that I think is probably not that interesting. I'm happy to agree on whatever convention for this thread.

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S.G. Janssens said:
It was about sequences of distinct points. (Then it is probably clear: Take any point ##x## in the space. An open nbh of ##x## is the complement of a finite set, and a sequence of distinct points leaves such a finite set eventually.)
Thanks, I was not aware of the topology being used.

S.G. Janssens said:
It was about sequences of distinct points. (Then it is probably clear: Take any point ##x## in the space. An open nbh of ##x## is the complement of a finite set, and a sequence of distinct points leaves such a finite set eventually.)
Or is ze risky ( Zariski ;) ) topology the same in the spaces in question as the finite complement topology?

S.G. Janssens
WWGD said:
Or is ze risky ( Zariski ;) ) topology the same in the spaces in question as the finite complement topology?
On the real line this is true, because there is a one-to-one correspondence between zero sets of polynomials and finite sets. Whether this is true in full generality, I do not know by heart. Probably @mathwonk knows.

WWGD
S.G. Janssens said:
On the real line this is true, because there is a one-to-one correspondence between zero sets of polynomials and finite sets. Whether this is true in full generality, I do not know by heart. Probably @mathwonk knows.
Actually, I was just thinking about it and the zero locus over 2+ dimensions can be ( uncountably) infinite, as in ##x^2+y^2+z^2-1## so likely the two topologies are not equal.

S.G. Janssens
WWGD said:
Actually, I was just thinking about it and the zero locus over 2+ dimensions can be ( uncountably) infinite, as in ##x^2+y^2+z^2-1## so likely the two topologies are not equal.
I don't think multivariates qualify, because if they would qualify, then I do not see how every sequence consisting of distinct points would have to converge to every point in the space:
mathwonk said:
You have made me realize another odd fact about the zariski topology that we use in algebraic geometry. On any affine space, say the real line, zariski closed sets are zero loci of polynomials, hence either the whole line, or finite sets. Hence every sequence consisting of distinct points, converges in this topology to every point of the line. Needless to say we don't use sequences much in this topology.

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WWGD
Very observant. Indeed what seems to be true in more variables is that any sequence of distinct points that eventually lies entirely on an algebraic curve, converges to every point of that curve.

mathwonk said:
Very observant. Indeed what seems to be true in more variables is that any sequence of distinct points that eventually lies entirely on an algebraic curve, converges to every point of that curve.

Suppose that ##C## and ##D## are two distinct algebraic curves in the (real) plane and let ##x \in C \setminus D##. If I understand you correctly, then any sequence of distinct points in ##C## should eventually be in ##C \setminus D##, for that sequence to converge to ##x##. However, what if ##C## and ##D## have a common component ##E## and the sequence in question lies entirely in ##E##?

I understand that the case of ##C## and ##D## having a common component is "non-generic", but the definition of convergence requires that it works in this case, too, doesn't it?

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