False statementConstruction: AI-adaptedVerification: AI-generatedprecheck passjudge pass (z-ai/glm-5.2)audited 2026-07-27

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Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
AI-adapted: a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
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FALSE: a function has at most one limit at every point of its domain, isolated points included

Statement

False claim: for every $A \subseteq R$ , every $f : A \to R$ and every $c \in A$ , at most one real $L$ satisfies

$(\forall ε > 0) (\exists δ > 0) (\forall x \in A) [0 < ∣ x - c ∣ < δ ⟹ ∣ f (x) - L ∣ < ε] . (*)$

Read the claim carefully: it is about the raw formula $(*)$ , extended to an arbitrary point $c$ of the domain. It is not a claim about The $ε$ - $δ$ limit $\lim_{x \to c} f (x) = L$ of $f : A \to R$ at a limit point $c$ of $A$ . That definition imposes $(*)$ only when $c$ is a limit point of $A$ (Limit point, isolated point, adherent point, derived set, and dense subset of $R$ ), and there at most one $L$ does satisfy it — that is exactly At a limit point of the domain a function has at most one limit, which is true and proved. The false claim is what one gets by deleting the limit-point requirement.

At an isolated point of $A$ the symbol $\lim_{x \to c} f (x)$ is undefined in this library, and the refutation below is the reason. If $c \in A$ is not a limit point of $A$ then some punctured neighbourhood of $c$ misses $A$ entirely (Limit point, isolated point, adherent point, derived set, and dense subset of $R$ ); the implication inside $(*)$ then has no instances at all for that $δ$ , so it holds vacuously, and it holds for every real $L$ at once. A formula satisfied by every real determines nothing, so no notation is introduced for it.

Facts & Assumptions

Given: The set $A : = {0} \cup [1, 2]$ (Intervals of $R$ : the nine order-convex forms, nondegeneracy, and length), the constant function $f : A \to R$ with $f (x) : = 0$ for every $x \in A$ , and the point $c : = 0 \in A$ .

[L1]

The $ε$ - $δ$ formula $(*)$ above, and the fact that The $ε$ - $δ$ limit $\lim_{x \to c} f (x) = L$ of $f : A \to R$ at a limit point $c$ of $A$ imposes it only at a limit point of the domain.

[L2]

Limit point and isolated point: $c$ is a limit point of $S$ when $N_{ε}^{*} (c) \cap S \neq \emptyset$ for every real $ε > 0$ , and $c \in S$ is an isolated point of $S$ when $N_{ε} (c) \cap S = {c}$ for some real $ε > 0$ ; for $c \in S$ these are exact opposites (Limit point, isolated point, adherent point, derived set, and dense subset of $R$ , The $ε$ -neighbourhood and the punctured $ε$ -neighbourhood of a point of $R$ ).

[L3]

Neighbourhoods: $N_{1} (0) = {y : ∣ y ∣ < 1}$ and $N_{1}^{*} (0) = {y : 0 < ∣ y ∣ < 1}$ (The $ε$ -neighbourhood and the punctured $ε$ -neighbourhood of a point of $R$ ).

[L4]

Absolute value and order: $∣ u ∣ \geq 0$ ; $∣ u ∣ = 0$ exactly when $u = 0$ ; $∣ u ∣ = u$ for $u \geq 0$ ; the order is total and trichotomy holds; and $0 < 1$ , so $0 \neq 1$ (Basic properties of the absolute value, The multiplicative identity is positive, Ordered field).

[L5]

Intervals: $[1, 2] = {y : 1 \leq y \leq 2}$ (Intervals of $R$ : the nine order-convex forms, nondegeneracy, and length).

Refutation

technique · direct

1.1

The point $0$ lies in $A$ , and $N_{1} (0) \cap A = {0}$ : an element of $A$ is either $0$ , which satisfies $∣ 0 ∣ = 0 < 1$ , or an element of $[1, 2]$ , which satisfies $∣ y ∣ = y \geq 1$ and so is not in $N_{1} (0)$ . Hence $0$ is an isolated point of $A$ and not a limit point of $A$ .

L2L3L4L5

1.2

The reals $0$ and $1$ are distinct.

2.1

Take $δ : = 1$ . No $x \in A$ satisfies $0 < ∣ x - 0 ∣ < 1$ : such an $x$ would lie in $N_{1}^{*} (0) \cap A$ , which is contained in $N_{1} (0) \cap A = {0}$ and excludes $0$ , hence is empty. So for every real $L$ and every real $ε > 0$ the choice $δ = 1$ makes the implication in $(*)$ vacuously true, and every real $L$ satisfies $(*)$ at $c = 0$ .

step 1.1L1L3L4

3.1

In particular $L = 0$ and $L = 1$ both satisfy $(*)$ at $c = 0$ , and they are distinct: more than one real satisfies the formula, so the claim is false.

step 1.2step 2.1L4∎

Remarks

This is the precise reason The $ε$ - $δ$ limit $\lim_{x \to c} f (x) = L$ of $f : A \to R$ at a limit point $c$ of $A$ carries the limit-point hypothesis. The hypothesis is not a convenience: it is what makes the quantified set $A \cap N_{δ}^{*} (c)$ nonempty for every $δ$ , and hence what makes $(*)$ capable of pinning $L$ down. With it, At a limit point of the domain a function has at most one limit proves uniqueness; without it, uniqueness is simply false, as above.
The true statement in the neighbourhood of the false one. For $c$ a limit point of $A$ : at most one $L$ satisfies $(*)$ — At a limit point of the domain a function has at most one limit. For $c$ an isolated point of $A$ : every $L$ satisfies $(*)$ , by the argument of step 2.1, which uses nothing about $f$ . So the dichotomy is total, and there is no intermediate case, because for $c \in A$ being isolated and being a limit point are exact opposites (Limit point, isolated point, adherent point, derived set, and dense subset of $R$ ).
Some texts do define the limit at an isolated point, declaring it to be $f (c)$ by fiat, so that "limit" and "continuity" coincide on such points. That is a convention, not a theorem, and this library declines it: a convention that assigns a value to an expression which the definition leaves underdetermined would have to be carried, and checked, in every later statement about limits. The companion page's counterexample exhibits the underdetermination concretely.

Depends on

Used by

On the domain {0} ∪ [1,2] every real is vacuously a limit at 0 Counterexample

Dependency tree · two levels

22 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

Limit of a function (Wikipedia) (standard reference, not scraped)
Isolated point (Wikipedia) (standard reference, not scraped)
J. Lebl, Basic Analysis I, §3.1 (standard reference, not scraped)