nLab
limit of a function

Limits of functions

Limits of functions

Idea

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

lim xcf(x)=L \lim_{x \to c} f(x) = L

means that f(x)f(x) is close to LL if xx is sufficiently close to cc. Of course, xx here is just a dummy variable, so this is really a ternary relation between ff, cc, and LL.

Definitions

Let XX and YY be topological spaces, let ff be a partial function from XX to YY (not assumed continuous or anything else), let cc be a limit point of the domain DD of ff in XX, and let LL be a point in YY.

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.

Definition

(French)

LL is a limit of ff approaching cc (assumed to be an adherent point of the domain DD) if, for each neighbourhood VV of LL, for some neighbourhood UU of cc, for each point xUDx \in U \cap D, we have f(x)Vf(x) \in V.

Definition

(English)

LL is a limit of ff approaching cc (assumed to be an accumulation point of the domain DD) if, for each neighbourhood VV of LL, for some punctured neighbourhood? UU of cc, for each point xUDx \in U \cap D, we have f(x)Vf(x) \in V. Equivalently, for each neighbourhood VV of LL, for some neighbourhood UU' of cc, for each point xUDx \in U' \cap D, if xcx \ne c, then f(x)Vf(x) \in V.

Note that in each case, UDU \cap D is inhabited precisely because cc is a limit point of DD. That is, in the French definition, UDU \cap D must be inhabited because an adherent point of DD is precisely a point cc such that each neighbourhood of cc meets DD; while in the English definition, UDU \cap D must be inhabited because an accumulation point of DD is precisely a point cc such that each punctured neighbourhood of cc meets DD. If we did not require cc to be such a limit point (in other words, if we allowed cc to have a [punctured] neighbourhood UU that was disjoint from DD), then every value LL would satisfy the definition vacuously.

The two notions of limit can each be defined in terms of the other:

Proposition

LL is a limit of ff approaching cc by the English definition if and only if LL is a limit of f| c^f|_{\hat{c}} approaching cc by the French definition, where f| c^f|_{\hat{c}} is the restriction of ff to the relative complement D{c}D \setminus \{c\} of cc in the original domain DD. Conversely, LL is a limit of ff approaching cc by the French definition if and only if cc is an adherent point of the domain DD and these two hypothetical conditions are met: if ff is defined at cc (in other words, if cc belongs to DD), then f(c)f(c) belongs to every neighbourhood of LL; and if cc is an accumulation point of DD, then LL is a limit of ff approaching cc by the English definition.

The basic difference is that the English definition doesn't care about f(c)f(c) itself, while the French definition does. For the French, ff must be continuous at cc if it is defined there, or equivalently (in the Hausdorff case), f(c)f(c) must equal the limit LL 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 DD, since we know what the value must be there. For cDc \notin D, then the two definitions are equivalent.

These limits can be defined as limits of a filter:

Proposition

Let 𝒩 c\mathcal{N}_c be the neighbourhood filter of cc, and let 𝒩˙ c\dot{\mathcal{N}}_c be the filter of punctured neighbourhoods of cc. Then the filter f(𝒩 c)f(\mathcal{N}_c) is proper iff cc is an adherent point of the domain DD, and f(𝒩˙ c)f(\dot{\mathcal{N}}_c) is proper iff cc is an accumulation point of DD. Futhermore, LL is a limit of ff approaching cc by the French definition iff LL is a limit of f(𝒩 c)f(\mathcal{N}_c), and LL is a limit of ff approaching cc by the English definition iff LL is a limit of f(𝒩˙ c)f(\dot{\mathcal{N}}_c).

Here, f()f(\mathcal{F}) is the filter generated by the filterbase of sets f(A)f(A) for AA \in \mathcal{F}, where f(A)f(A) is the image {f(x)|xA}\{ f(x) \;|\; x \in A \}. As an immediate corollary, if YY is Hausdorff, then LL must be the only limit of ff approaching cc.

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 XX and YY be topological spaces, let RR be a relation from XX to YY, let cc be a limit (adherent or accumulation) point of the domain DD of RR in XX, and let LL be a point in YY.

Definition

LL is a limit of RR approaching cc if, for each neighbourhood VV of LL, for some [punctured] neighbourhood UU of cc, for each point xUDx \in U \cap D, we have R(x)VR(x) \subseteq V.

Here, R(x)R(x) is the set {yY|(x,y)R}\{ y \in Y \;|\; (x,y) \in R \} of values of RR at xx. Again, there are both French-style and English-style versions of the definition. This can also be defined as the limit of a filter R(𝒩 c)R(\mathcal{N}_c) or R(𝒩˙ c)R(\dot{\mathcal{N}}_c), where R()R(\mathcal{F}) is generated by the sets R(A)= xAR(x)R(A) = \bigcup_{x \in A} R(x) for AA \in \mathcal{F}. In particular, limits of multivalued functions are still unique, so long as the target YY is Hausdorff.

We may generalize further from relations to spans. Let XX and YY be topological spaces, let Γ\Gamma be any set, let g:ΓXg\colon \Gamma \to X and f:ΓYf\colon \Gamma \to Y be (total) functions, let cc be a limit (adherent or accumulation) point of the range DD of gg in XX, and let LL be a point in YY.

Definition

LL is a limit of the span (g,f)(g,f) approaching cc if, for each neighbourhood VV of LL, for some [punctured] neighbourhood UU of cc, for each element γ\gamma of the preimage g *(UD)g^*(U \cap D), we have f(γ)Vf(\gamma) \in V.

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 f(g *(𝒩 c))f(g^*(\mathcal{N}_c)) or f(g *(𝒩˙ c))f(g^*(\dot{\mathcal{N}}_c)), where g *()g^*(\mathcal{F}) consists of the preimages g *(A)g^*(A) for AA \in \mathcal{F}. In particular, limits of spans to a Hausdorff space are unique.

Limits of spans are no more general than limits of relations:

Proposition

LL is a limit of the span (g,f)(g,f) approaching cc if and only if LL is a limit of the range ran(g,f)={g(γ),f(γ)|γΓ}\ran(g,f) = \{ g(\gamma), f(\gamma) \;|\; \gamma \in \Gamma \} approaching cc.

(But often the span is more convenient to refer to than its range.)

Finally, we may impose restrictions on the limit:

Definition

Let CC be a subset of XX, and suppose that cc is a limit (adherent or accumulation) point of CC. Then LL is a limit of the function ff (or the relation RR, or the span (g,f)(g,f)) approaching cc in CC if LL is a limit of the restriction f| Cf|_C of ff to DCD \cap C (or the restriction R(C×Y)R \cap (C \times Y) of RR, or the restriction (g| g *(C),f| g *(C))(g|_{g^*(C)},f|_{g^*(C)}) of (g,f)(g,f)). In the case of the limit of a span, we can also let CC be a subset of Γ\Gamma; then the limit in question is the limit of (g| C,f| C)(g|_C,f|_C).

Yet again, there are both French-style and English-style versions, although these are equivalent when cCc \notin C. 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:

Proposition

LL is a limit (in the English sense) of ff (or RR, or (g,f)(g,f)) approaching cc in CC if and only if LL is a limit (in the French sense) of ff (or RR, or (g,f)(g,f)) approaching cc in CcC \setminus c.

To go more general than this would seem to require referring directly to a net ν\nu or filter \mathcal{F} on XX, at which point we may as well just talk about the limit of the net fνf \circ \nu or the limit of the filter f()f(\mathcal{F}) (or of the filter R()R(\mathcal{F}) or of the filter f(g *())f(g^*(\mathcal{F}))).

Notation

In the Hausdorff case, we usually write

L=lim xcf(x) L = \lim_{x \to c} f(x)

to say that LL is a limit, or rather the limit, of ff approaching cc. Or indeed, just write

lim xcf(x) \lim_{x \to c} f(x)

for the limit, if it exists (so, like 1/x1/x when xx is an arbitrary real number, this is a symbol for a thing that might not be defined). Since xx is just a dummy variable here, one can try a notation that does not refer to it, such as lim cf \lim_c f or f(c ±)f(c^\pm).

The last notation is not very common, but some variations of it are when XX is a topological poset: f(c )f(c^-) is the limit of ff approaching cc in C={xX|x<c}C = \{ x \in X \;|\; x \lt c \}, while f(c +)f(c^+) is the limit of ff approaching cc in C={xX|x>c}C = \{ x \in X \;|\; x \gt c \}. Similarly, f()f(\infty) is the limit of ff approaching \infty, where the domain XX is taken to be the disjoint union (as a set) of the original topological poset and a point \infty that is greater than every original element and whose neighbourhoods are the upper sets of XX; while f()f(-\infty) is the limit of ff approaching -\infty, where the domain XX is taken to be the disjoint union of the original topological poset and a point -\infty that is less than every original element and whose neighbourhoods are the lower sets of XX. (In all of these, the French and English definitions agree.)

For the general restricted case, write

lim xccCf(x) \lim_{x \to c \atop c \in C} f(x)

for the limit of ff approaching cc in CC. The dummy variable is quite useful here, since neither ff nor CC have to be given names but can be given by formulas instead. (For example, f(c +)f(c^+) is lim xcx>cf(x)\lim_{x \to c \atop x \gt c} f(x).)

In the non-Hausdorff case, we can use the same notation but interpret it as referring to a subset of YY instead of an element of YY. Then this subset always exists; it just might be empty. Sometimes a capitalized LimLim is used to emphasize that this is now a set. This set could also be taken to be all of YY whenever cc 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 YY is non-Hausdorff, but common even in the Hausdorff case, is

f(x)xcL f(x) \underset{x \to c}{\to} L

to mean that LL is a limit of ff approaching cc. Again, we can add cCc \in C if we wish to take the limit in CC. And again, the dummy variable means that we don't need a name for ff (or CC), just a formula. Often one writes only f(x)Lf(x) \to L and puts xcx \to c (and cCc \in C if appropriate) off to the side somewhere:

  • f(x)Lf(x) \to L as xcx \to c while cCc \in C,

or with more words: f(x)f(x) approaches LL as xx approaches cc while cCc \in C.

Most of this notation can also be used for limits of spans:

L=lim g(γ)cγCf(γ), LLimg(γ)cγCf(γ), f(γ)g(γ)cγCL \array { L = \lim_{g(\gamma) \to c \atop \gamma \in C} f(\gamma) ,\\ L \in \underset{g(\gamma) \to c \atop \gamma \in C}{Lim} f(\gamma) ,\\ f(\gamma) \underset{g(\gamma) \to c \atop \gamma \in C}{\to} L }

all mean that LL is a limit of (g,f)(g,f) approaching cc in CC. Thus the most general notion of limit appearing here is to say that f(γ)f(\gamma) approaches LL as g(γ)g(\gamma) approaches cc while γC\gamma \in C.

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 xcx \ne c 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.

Examples

A function ff is continuous at a point cc in its domain DD, if and only if f(c)f(c) is a limit of ff approaching cc by the French definition; ff is continuous at cc if and only if, if cc is an accumulation point of DD, then f(c)f(c) is a limit of ff approaching cc by the English definition. (If cc is an isolated point of DD, then of course ff is continuous there.)

Given a real-valued function FF defined on a real interval [a,b][a,b], we can speak of tagged partitions of [a,b][a,b] and consider the Riemann sums? of FF on these tagged partitions. Let XX and YY each be the real line, let Γ\Gamma be the set of tagged partitions of [a,b][a,b], let g:ΓXg\colon \Gamma \to X map a partition to the maximum length of its parts (often called the norm or mesh of the partition), and let f:ΓYf\colon \Gamma \to Y map a tagged partition to the Riemann sum of FF on it. Then the Riemann integral of FF on [a,b][a,b] is defined to be the limit of the span (f,g)(f,g) approaching 00. 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 AA be a directed set and let ν:AY\nu\colon A \to Y be a net in YY. Give AA the discrete topology, and let XX be the disjoint union (as a set) of AA and {}\{\infty\}, made into a proset with x<x \lt \infty for xAx \in A, and made into a topological space with the neighbourhoods of \infty being the upper sets of XX. Interpret ν\nu as a partial function from XX to YY. Then the limits of ν\nu approaching \infty are precisely the limits of ν\nu as a net.

Properties

Some basic relationships between the definitions are in the definitions section.

As stated in the examples, ff is continuous at a point cc in its domain DD if and only if f(c)f(c) is a limit of ff approaching cc by the French definition, while ff is continuous at cc if and only if, if cc is an accumulation point of DD, then f(c)f(c) is a limit of ff approaching cc 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 LL is a limit of ff at cc if setting f(c)f(c) to LL makes ff continuous. For once, this is easier to describe using the English definition: LL is the limit of ff at an accumulation point cc of the domain DD if and only if f cLf_{c \mapsto L} is continuous at cc, where

f cL(x){f(x) xc,xD, L x=c. f_{c \mapsto L}(x) \coloneqq \left\{ \array{ f(x) & x \ne c,\; x \in D ,\\ L & x = c .}\right.

For the French definition, in addition to allowing cc to be any adherent point of DD, we use

f cL(x){f(x) xD, L x=c, f_{c \mapsto L}(x) \coloneqq \left\{ \array{ f(x) & x \in D ,\\ L & x = c ,}\right.

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 YY of ff is Hausdorff (or at least T 1\mathrm{T}_1); 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 fgf \circ g approaching cc is equal to the limit of ff approaching the limit of gg approaching cc, if the latter exists:

lim xcf(g(x))lim ulim xcg(x)f(u), \lim_{x \to c} f(g(x)) \risingdotseq \lim_{u \to \lim_{x \to c} g(x)} f(u) ,

or equivalently (fg)(c ±)f(g(c ±) ±)(f \circ g)(c^\pm) \risingdotseq f(g(c^\pm)^\pm), where ABA \risingdotseq B means that AA and BB are equal if BB exists (and ABA \fallingdotseq B means that AA and BB are equal if AA exists). Using the English definition, this holds if we impose the requirement that if gg takes the value lim cg\lim_c g arbitrarily close to cc (regardless of its value at cc) and ff is defined there, then ff is continuous there. Either way, we can write

lim xcf(g(x))=f(lim xcg(x)) \lim_{x \to c} f(g(x)) = f\Big(\lim_{x \to c} g(x)\Big)

if ff is continuous at lim cg\lim_c g.

Last revised on June 28, 2020 at 22:56:19. See the history of this page for a list of all contributions to it.