Although it has less emphasis in advanced mathematics, and although its definition is more complicated than that of the limit of a sequence, the first concept of limit seen by most students is that of a limit of a function. This is the idea that
means that is close to if is sufficiently close to . Of course, here is just a dummy variable, so this is really a ternary relation between , , and .
Let and be topological spaces, let be a partial function from to (not assumed continuous or anything else), let be a limit point of the domain of in , and let be a point in .
There are actually two definitions of ‘limit point’ in the literature: an adherent point and an accumulation point. And there are two definitions of ‘limit’ in this context: the usual French-language one (following Bourbaki) and the usual English-language one. These definitions correspond respectively. In both cases, the first definition is simpler, while the second is more common.
(French)
is a limit of approaching (assumed to be an adherent point of the domain ) if, for each neighbourhood of , for some neighbourhood of , for each point , we have .
(English)
is a limit of approaching (assumed to be an accumulation point of the domain ) if, for each neighbourhood of , for some punctured neighbourhood? of , for each point , we have . Equivalently, for each neighbourhood of , for some neighbourhood of , for each point , if , then .
Note that in each case, is inhabited precisely because is a limit point of . That is, in the French definition, must be inhabited because an adherent point of is precisely a point such that each neighbourhood of meets ; while in the English definition, must be inhabited because an accumulation point of is precisely a point such that each punctured neighbourhood of meets . If we did not require to be such a limit point (in other words, if we allowed to have a [punctured] neighbourhood that was disjoint from ), then every value would satisfy the definition vacuously.
The two notions of limit can each be defined in terms of the other:
is a limit of approaching by the English definition if and only if is a limit of approaching by the French definition, where is the restriction of to the relative complement of in the original domain . Conversely, is a limit of approaching by the French definition if and only if is an adherent point of the domain and these two hypothetical conditions are met: if is defined at (in other words, if belongs to ), then belongs to every neighbourhood of ; and if is an accumulation point of , then is a limit of approaching by the English definition.
The basic difference is that the English definition doesn't care about itself, while the French definition does. For the French, must be continuous at if it is defined there, or equivalently (in the Hausdorff case), must equal the limit if it exists at all. This makes the French definition more strict when both make sense, but it also allows the French definition to make sense at isolated points of the domain , since we know what the value must be there. For , then the two definitions are equivalent.
These limits can be defined as limits of a filter:
Let be the neighbourhood filter of , and let be the filter of punctured neighbourhoods of . Then the filter is proper iff is an adherent point of the domain , and is proper iff is an accumulation point of . Futhermore, is a limit of approaching by the French definition iff is a limit of , and is a limit of approaching by the English definition iff is a limit of .
Here, is the filter generated by the filterbase of sets for , where is the image . As an immediate corollary, if is Hausdorff, then must be the only limit of approaching .
We can generalize from functions to multivalued functions; since we're already using partial functions, this means that we are dealing with an arbitrary binary relation. That is, let and be topological spaces, let be a relation from to , let be a limit (adherent or accumulation) point of the domain of in , and let be a point in .
is a limit of approaching if, for each neighbourhood of , for some [punctured] neighbourhood of , for each point , we have .
Here, is the set of values of at . Again, there are both French-style and English-style versions of the definition. This can also be defined as the limit of a filter or , where is generated by the sets for . In particular, limits of multivalued functions are still unique, so long as the target is Hausdorff.
We may generalize further from relations to spans. Let and be topological spaces, let be any set, let and be (total) functions, let be a limit (adherent or accumulation) point of the range of in , and let be a point in .
is a limit of the span approaching if, for each neighbourhood of , for some [punctured] neighbourhood of , for each element of the preimage , we have .
Once again, there are both French-style and English-style versions of the definition. And this too can be defined as the limit of a filter or , where consists of the preimages for . In particular, limits of spans to a Hausdorff space are unique.
Limits of spans are no more general than limits of relations:
is a limit of the span approaching if and only if is a limit of the range approaching .
(But often the span is more convenient to refer to than its range.)
Finally, we may impose restrictions on the limit:
Let be a subset of , and suppose that is a limit (adherent or accumulation) point of . Then is a limit of the function (or the relation , or the span ) approaching in if is a limit of the restriction of to (or the restriction of , or the restriction of ). In the case of the limit of a span, we can also let be a subset of ; then the limit in question is the limit of .
Yet again, there are both French-style and English-style versions, although these are equivalent when . And once more, this is the limit of a filter, since it reduces to a previous definition.
This concept allows us to define the English version of limit directly as a French limit:
is a limit (in the English sense) of (or , or ) approaching in if and only if is a limit (in the French sense) of (or , or ) approaching in .
To go more general than this would seem to require referring directly to a net or filter on , at which point we may as well just talk about the limit of the net or the limit of the filter (or of the filter or of the filter ).
In the Hausdorff case, we usually write
to say that is a limit, or rather the limit, of approaching . Or indeed, just write
for the limit, if it exists (so, like when is an arbitrary real number, this is a symbol for a thing that might not be defined). Since is just a dummy variable here, one can try a notation that does not refer to it, such as or .
The last notation is not very common, but some variations of it are when is a topological poset: is the limit of approaching in , while is the limit of approaching in . Similarly, is the limit of approaching , where the domain is taken to be the disjoint union (as a set) of the original topological poset and a point that is greater than every original element and whose neighbourhoods are the upper sets of ; while is the limit of approaching , where the domain is taken to be the disjoint union of the original topological poset and a point that is less than every original element and whose neighbourhoods are the lower sets of . (In all of these, the French and English definitions agree.)
For the general restricted case, write
for the limit of approaching in . The dummy variable is quite useful here, since neither nor have to be given names but can be given by formulas instead. (For example, is .)
In the non-Hausdorff case, we can use the same notation but interpret it as referring to a subset of instead of an element of . Then this subset always exists; it just might be empty. Sometimes a capitalized is used to emphasize that this is now a set. This set could also be taken to be all of whenever is not a limit point of the appropriate kind, but this is likely to lead to confusion if not explicitly warned about.
Another notation, especially useful when is non-Hausdorff, but common even in the Hausdorff case, is
to mean that is a limit of approaching . Again, we can add if we wish to take the limit in . And again, the dummy variable means that we don't need a name for (or ), just a formula. Often one writes only and puts (and if appropriate) off to the side somewhere:
or with more words: approaches as approaches while .
Most of this notation can also be used for limits of spans:
all mean that is a limit of approaching in . Thus the most general notion of limit appearing here is to say that approaches as approaches while .
It would be nice to have notation to distinguish the French and English versions of limit. One way would be to adopt the French definition by default and add the restriction to produce the English version. Unfortunately, the English definition is the default in most of the world, and then there is no slick way to denote the French version.
A function is continuous at a point in its domain , if and only if is a limit of approaching by the French definition; is continuous at if and only if, if is an accumulation point of , then is a limit of approaching by the English definition. (If is an isolated point of , then of course is continuous there.)
Given a real-valued function defined on a real interval , we can speak of tagged partitions of and consider the Riemann sums? of on these tagged partitions. Let and each be the real line, let be the set of tagged partitions of , let map a partition to the maximum length of its parts (often called the norm or mesh of the partition), and let map a tagged partition to the Riemann sum of on it. Then the Riemann integral of on is defined to be the limit of the span approaching . That is, the Riemann integral of a function on an interval is the limit of the Riemann sum of the function on a tagged partition of the interval as the mesh of the partition approaches zero.
Let be a directed set and let be a net in . Give the discrete topology, and let be the disjoint union (as a set) of and , made into a proset with for , and made into a topological space with the neighbourhoods of being the upper sets of . Interpret as a partial function from to . Then the limits of approaching are precisely the limits of as a net. That is, .
Some basic relationships between the definitions are in the definitions section.
As stated in the examples, is continuous at a point in its domain if and only if is a limit of approaching by the French definition, while is continuous at if and only if, if is an accumulation point of , then is a limit of approaching by the English definition. In this way, pointwise continuity may be defined using limits.
Conversely, limits can be defined using pointwise continuity; the basic idea is that is a limit of at if setting to makes continuous. For once, this is easier to describe using the English definition: is the limit of at an accumulation point of the domain if and only if is continuous at , where
For the French definition, in addition to allowing to be any adherent point of , we use
where the definition by cases is interpreted to mean that the value of the function is undefined if the two cases overlap and give different results. Except that even this is only correct when the target of is Hausdorff (or at least ); in general, we have to allow the value to be defined if the two cases overlap and give different results as long as one result is in the closure of the other (and then we use the other).
There is a Chain Rule for limits. Using the French definition, the limit of a composite function approaching is equal to the limit of approaching the limit of approaching , if the latter exists:
or equivalently , where means that and are equal if exists (and means that and are equal if exists). Using the English definition, this holds if we impose the requirement that if takes the value arbitrarily close to (regardless of its value at ) and is defined there, then is continuous there. Either way, we can write
if is continuous at .
Last revised on January 14, 2023 at 20:35:08. See the history of this page for a list of all contributions to it.