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

How statement and proof provenance work

The first chip identifies the source of the statement or construction; the second identifies the source of its local proof or verification.

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.
AI-generated: a genuinely novel statement formulated by AI, with no source for the claim itself.

These labels describe origin, not correctness: citations and verification chips remain separate evidence.

FALSE: $\lim_{x \to c} f (x) = f (c)$ whenever both sides exist

Statement

False claim: if $A \subseteq R$ , if $f : A \to R$ , if $c \in A$ is a limit point of $A$ (Limit point, isolated point, adherent point, derived set, and dense subset of $R$ ) and if the limit of $f$ at $c$ exists (The $ε$ - $δ$ limit $\lim_{x \to c} f (x) = L$ of $f : A \to R$ at a limit point $c$ of $A$ ), then

$\lim_{x \to c} f (x) = f (c) .$

Both sides of the asserted equation are defined under the stated hypotheses: the left because the limit is assumed to exist and is single valued (At a limit point of the domain a function has at most one limit), the right because $c \in A$ . The claim is that they always agree, and that is false.

Why it is tempting. The condition $∣ f (x) - L ∣ < ε$ is imposed on points $x$ arbitrarily close to $c$ , and it feels as though $x = c$ were the limiting case of that. It is not: The $ε$ - $δ$ limit $\lim_{x \to c} f (x) = L$ of $f : A \to R$ at a limit point $c$ of $A$ quantifies over $0 < ∣ x - c ∣ < δ$ , and the strict inequality on the left removes $x = c$ from the quantifier entirely. Changing the value of $f$ at the single point $c$ therefore changes nothing on the left-hand side and everything on the right.

What is true. The equation above is not a theorem but a condition, and it is the condition the next page of this track takes as the definition of continuity at $c$ . This library states it as a hypothesis and never as a consequence; hypothesis (i) of Composition of limits holds under either hypothesis: $f$ is defined at $L$ with value $M$ , or $g$ avoids $L$ on a punctured neighbourhood of $c$ is exactly this condition for the outer function.

Facts & Assumptions

Given: The set $A : = R$ , the point $c : = 0$ , and the function $f : R \to R$ defined by $f (x) : = 0$ for $x \neq 0$ and $f (0) : = 1$ .

[L1]

The limit condition: $\lim_{x \to c} h (x) = L$ means that for every real $ε > 0$ there is a real $δ > 0$ such that every $x$ in the domain of $h$ with $0 < ∣ x - c ∣ < δ$ satisfies $∣ h (x) - L ∣ < ε$ (The $ε$ - $δ$ limit $\lim_{x \to c} f (x) = L$ of $f : A \to R$ at a limit point $c$ of $A$ ).

[L2]

Limit point: $c$ is a limit point of $S$ when every punctured neighbourhood $N_{ε}^{*} (c)$ meets $S$ ; and punctured neighbourhoods in $R$ are never empty (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]

Absolute value: $∣ 0 ∣ = 0$ , and $∣ u ∣ \geq 0$ (Basic properties of the absolute value).

[L4]

Order in $R$ : trichotomy, so every real either equals $0$ or does not, and never both; and $0 < 1$ , so $1 \neq 0$ (The multiplicative identity is positive, Ordered field).

Refutation

technique · direct

1.1

The point $c = 0$ lies in $A = R$ and is a limit point of $R$ : for every real $ε > 0$ the punctured neighbourhood $N_{ε}^{*} (0)$ is nonempty and is contained in $R$ , so it meets $R$ .

1.2

$f$ is a well-defined function on $R$ , since by trichotomy every real either equals $0$ or does not, exclusively; and the reals $0$ and $1$ are distinct.

2.1

The limit of $f$ at $0$ exists and equals $0$ : given an arbitrary real $ε > 0$ , take $δ : = 1 > 0$ ; every $x \in R$ with $0 < ∣ x - 0 ∣ < 1$ has $∣ x ∣ \neq 0$ , hence $x \neq 0$ , hence $f (x) = 0$ and $∣ f (x) - 0 ∣ = ∣ 0 ∣ = 0 < ε$ .

step 1.1step 1.2L1L3L4

3.1

Yet $f (0) = 1$ , and $1 \neq 0 = \lim_{x \to 0} f (x)$ . So at the point $c = 0$ of the domain, which is a limit point of the domain, the limit exists and differs from the value: the claim is false.

step 1.2step 2.1L4∎

Remarks

The witness is the smallest possible one. It differs from a constant function at exactly one point, and the limit cannot see that point. Any function agreeing with a constant off $c$ and taking a different value at $c$ would serve equally well; the companion page works this witness out in full, computes its one-sided limits, and shows that redefining the single value repairs the equality.
Where the false claim does hold. Under the extra hypothesis that $\lim_{x \to c} f (x) = f (c)$ — which is what continuity at $c$ will mean — it holds trivially, and that is the only sense in which it is ever true. It is emphatically not a consequence of the limit existing.
The consequence for composition. Because $f (c)$ is invisible to the limit, substituting an inner function that takes the value $c$ is not licensed by the limits alone; that is the content of FALSE: $\lim_{x \to c} f (g (x)) = M$ whenever $\lim_{x \to c} g = L$ and $\lim_{y \to L} f = M$ , whose witness is built from this one.

Depends on

Used by

The function equal to 0 off the origin and to 1 at the origin has limit 0 ≠ 1 there Counterexample
FALSE: lim_x → c f(g(x)) = M whenever lim_x → c g = L and lim_y → L f = M False statement

Dependency tree · two levels

21 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

J. Lebl, Basic Analysis I, §3.1: Limits of functions (standard reference, not scraped)
Limit of a function (Wikipedia) (standard reference, not scraped)
Classification of discontinuities (Wikipedia) (standard reference, not scraped)
J. Lebl, Basic Analysis I, §3.1 (standard reference, not scraped)

FALSE: lim⁡x→cf(x)=f(c)limx→c​f(x)=f(c) whenever both sides exist