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CLRg property

A common-limit-in-the-range condition for self-maps f and g: some sequence x_k has both f(x_k) and g(x_k) converge to the same point g(x), placing the shared limit in the range of g without requiring that range to be closed.

Version
v1 · 2026-09-28 · History
Domain-specific #
8506
Domain group
Formal Sciences
Origin domain
Mathematics
Subdomains
Fixed Point Theory, Metric Spaces → Mathematics

Core Idea

The CLRg property—common limit in the range of g—is an existential convergence condition on two self-maps. A pair f,g satisfies it if some sequence x_k has f(x_k) and g(x_k) converging to the same point, and that point is g(x) for some x in the space.

The attained-range clause is the point of the definition. It allows fixed-point arguments without assuming that the whole range g(X) is closed: only the relevant shared limit must be realized by g. CLRg alone is not a fixed-point theorem and does not say f(x)=g(x)=x.

How would you explain it like I'm…

 

No faithful explanation at this level. Two of three generators judged eli5 unreachable: a five-year-old picture of two things heading to the same spot collapses into them meeting at a common or fixed point, exactly what CLRg does not assert; the property only concerns a shared limit of a sequence that lies in g's range. A's lone two-robots version was the minority.

Common Limit Reached by g

The CLRg property is about two functions, f and g, that take points of a space to points of the same space. It asks for a sequence of inputs where the outputs of f and the outputs of g both get closer and closer to one common point. On top of that, this common point must actually be an output of g, meaning g(x) equals it for some input x. The name stands for 'common limit in the range of g'. It is a helpful step in some proofs about fixed points, but by itself it does not prove that f and g agree anywhere.

Common Limit in the Range of g

The CLRg property, short for 'common limit in the range of g', is a condition on two self-maps f and g of a space X. The pair satisfies it if there is a sequence x_k such that f(x_k) and g(x_k) both converge to the same point t, and that point is actually of the form g(x) for some x in X. The key part is the second condition, that the limit lies in the range of g. It lets fixed-point arguments go through without assuming the whole image g(X) is a closed set; only the particular common limit needs to be hit by g. CLRg by itself is not a fixed-point theorem: it does not claim f(x) = g(x) = x for any x, and further conditions are needed to reach that conclusion.

 

The CLRg property (common limit in the range of g) is an existential convergence condition on a pair of self-maps f, g of a space X: there exists a sequence (x_k) in X with lim f(x_k) = lim g(x_k) = g(x) for some x ∈ X. The attained-range clause is the substance of the definition. Fixed-point arguments often need the common limit to be realized by g; rather than assuming the whole range g(X) is closed, CLRg asks only that the relevant shared limit lie in g(X). It is therefore a weaker, targeted hypothesis rather than a global closedness assumption. CLRg by itself is not a fixed-point theorem: it does not assert that f(x) = g(x), let alone f(x) = g(x) = x; further hypotheses are needed to reach such conclusions.

Structural Signature

Sig role-phrases:

  • metric-space carrier. Provides a nonempty domain and convergence relation. Constitutive setting. If altered: Without convergence structure the condition is undefined.
  • self-maps f and g. Generate two image sequences from the same inputs. Constitutive operators. If altered: One map alone cannot express the common-limit relation.
  • witness sequence. Supplies points x_k whose images are compared. Existential witness. If altered: The property need not hold for every sequence.
  • common limit. Makes both image sequences converge to one value. Identity-bearing relation. If altered: Different limits fail the property.
  • range witness g(x). Requires the common limit to be attained by g. Distinctive range clause. If altered: A shared limit merely in the closure of g(X) is insufficient.

What It Is Not

  • Common fixed point. Does the limit point itself satisfy both maps?
  • Closed range. Is only one limit attained or the whole range closed?
  • Coincidence point. Is f(x)=g(x) asserted at one input?
  • Compatible maps. Is a commutation-style condition being substituted?

Scope of Application

Use CLRg only after naming the two maps, a witness sequence, both image limits, and the point whose g-image attains the limit.

  • Fixed-point theory. Supplies a convergence hypothesis.
  • Metric spaces. Provides sequence convergence.
  • Fuzzy metric work. Extends contractive-map arguments in the cited context.
  • Nonclosed ranges. Avoids a global closedness assumption.
  • Map compatibility studies. Compares distinct hypotheses carefully.

Clarity

Convergence of f(x_k) and g(x_k) to one point is not enough if that point is merely approached by g-values. Writing it explicitly as g(x) verifies range attainment.

Manages Complexity

The property compresses four existential choices—maps, sequence, point, and limit—into a reusable hypothesis. It guarantees none of uniqueness, contraction, or fixedness by itself.

Abstract Reasoning

  1. Specify the carrier and convergence notion.
  2. Name f and g as self-maps.
  3. Exhibit a single common witness sequence.
  4. Compute both image-sequence limits.
  5. Show the common value equals g(x) for an actual x.

Knowledge Transfer

Common-limit-plus-attainment reasoning transfers to operator and approximation arguments, but the CLRg name and designated range of g are fixed-point-theory specific. The nearest stopping boundary is explicit: A common limit in the closure of g(X) is closest: CLRg additionally requires that the limit equal an actual value g(x). The inclusion test remains: The pair (f,g) has CLRg when there exist x_k and x such that f(x_k) and g(x_k) both converge to the attained value g(x). The structure no longer applies when the case exits when the image limits differ, convergence fails, or the shared limit is not in g's range.

Examples

Canonical

On [0,infinity), f(x)=x/4 and g(x)=3x/4 with x_k=1/k give f(x_k)→0 and g(x_k)→0=g(0).

Mapped back: metric-space carrier → nonnegative reals; self-maps f and g → quarter and three-quarter scaling; witness sequence → 1/k; common limit → 0; range witness g(x) → g(0)=0.

Applied / In Practice

If f(x_k) and g(x_k) both approach L but no x satisfies g(x)=L, the pair has a shared closure limit but fails CLRg.

Mapped back: metric-space carrier → given metric space; self-maps f and g → defined maps; witness sequence → shared sequence; common limit → L; range witness g(x) → absent.

Structural Tensions

T1: weak global assumption vs. strong local witness. No closed range is required, but the one relevant limit must be attained. Diagnostic: Where is the range witness?

T2: existence vs. constructibility. The property asserts a sequence and point without necessarily supplying an algorithm to find them. Diagnostic: Is the witness explicit or only proved?

Structural–Framed Character

Description turns on metric-space carrier, self-maps f and g, witness sequence, common limit, range witness g(x). Skeletal core. Two transformations share a limiting output that one transformation actually attains. Domain-bound accent. Self-maps, metric convergence, image ranges, and fixed-point hypotheses define CLRg. Transfer remains bounded because Why not prime. Limit-attainment structure is portable; CLRg is a specialized named property. The negative boundary is concrete: Any coincidence point, compatible maps, shared accumulation point, convergent image sequence, fixed point, or closed range is not automatically the CLRg property. CLRg is structural-formal: its sequence, convergence, equality, and range clauses are mathematically exact. Its character: a shared image limit anchored inside a designated map's range.

Structural Core vs. Domain Accent

Skeletal core. Two transformations share a limiting output that one transformation actually attains.

Domain-bound accent. Self-maps, metric convergence, image ranges, and fixed-point hypotheses define CLRg.

Why not prime. Limit-attainment structure is portable; CLRg is a specialized named property.

  • Convergence. Both image sequences approach one point.
  • Attainment. The point lies in g's actual range.
  • No strict parent is asserted.

Neighborhood in Abstraction Space

CLRg property sits in a moderately populated region (51st percentile for distinctiveness): it has near-neighbors but no dense thicket of look-alikes.

Family — Abstract Algebra & Category Theory (24 abstractions)

Nearest neighbors

Computed from structural-signature embeddings · 2026-10-08

Not to Be Confused With

  • Common fixed point. Tell: Does the limit point itself satisfy both maps?
  • Closed range. Tell: Is only one limit attained or the whole range closed?
  • Coincidence point. Tell: Is f(x)=g(x) asserted at one input?
  • Compatible maps. Tell: Is a commutation-style condition being substituted?

References

  • Frozen Wikipedia discovery revision: https://en.wikipedia.org/wiki/CLRg_property (revision 1339201393).

The frozen Wikipedia revision is discovery provenance. The retained source set was reviewed for identity, formal or operational relation, and scope. The encyclopedia's structural synthesis is bounded to those claims; a thin authority surface is recorded as a nonblocking source-strengthening repair rather than concealed.