False statementConstruction: AI-adaptedVerification: AI-generatedSession-authored (Fable 5 assisted)precheck 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 $A \subseteq \mathbb{R}$ , if $f : A \to \mathbb{R}$ , if $c \in A$ is a limit point of $A$ (Limit point, isolated point, adherent point, derived set, and dense subset of $\mathbb{R}$ ) and if the limit of $f$ at $c$ exists (The $\varepsilon$ - $\delta$ limit $\lim_{x \to c} f(x) = L$ of $f : A \to \mathbb{R}$ 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 $|f(x) - L| < \varepsilon$ 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 $\varepsilon$ - $\delta$ limit $\lim_{x \to c} f(x) = L$ of $f : A \to \mathbb{R}$ at a limit point $c$ of $A$ quantifies over $0 < |x - c| < \delta$ , 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 $A := \mathbb{R}$ , the point $c := 0$ , and the function $f : \mathbb{R} \to \mathbb{R}$ defined by $f(x) := 0$ for $x \ne 0$ and $f(0) := 1$ .

[L1]

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

[L2]

Limit point: $c$ is a limit point of $S$ when every punctured neighbourhood $N^{*}_{\varepsilon}(c)$ meets $S$ ; and punctured neighbourhoods in $\mathbb{R}$ are never empty (Limit point, isolated point, adherent point, derived set, and dense subset of $\mathbb{R}$ , The $\varepsilon$ -neighbourhood and the punctured $\varepsilon$ -neighbourhood of a point of $\mathbb{R}$ ).

[L3]

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

[L4]

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

Refutation

technique · direct

1.1

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

1.2

$f$ is a well-defined function on $\mathbb{R}$ , 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 $\varepsilon > 0$ , take $\delta := 1 > 0$ ; every $x \in \mathbb{R}$ with $0 < |x - 0| < 1$ has $|x| \ne 0$ , hence $x \ne 0$ , hence $f(x) = 0$ and $|f(x) - 0| = |0| = 0 < \varepsilon$ .

step 1.1step 1.2L1L3L4

3.1

Yet $f(0) = 1$ , and $1 \ne 0 = \lim_{x \to 0} f(x)$ . 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 · next 3 levels

Direct dependencies and their dependencies through the next three levels: 33 results over 13 levels. An arrow runs from a result to what uses it, and this result sits at the bottom with a heavier outline. Click 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)\lim_{x \to c} f(x) = f(c)limx→c​f(x)=f(c) whenever both sides exist