TheoremStatement: AI-adaptedProof: AI-generatedprecheck passverified 2026-08-09 (gpt-5.6-terra-codex-subscription)

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.

Composition of limits holds under either hypothesis: $f$ is defined at $L$ with value $M$ , or $g$ avoids $L$ on a punctured neighbourhood of $c$

Statement

Let $A, B \subseteq R$ , let $g : A \to R$ with $g (A) \subseteq B$ , and let $f : B \to R$ , so that the composite $f \circ g : A \to R$ is defined. Let $c$ be a limit point of $A$ and $L$ a limit point of $B$ (Limit point, isolated point, adherent point, derived set, and dense subset of $R$ ), and suppose the limits

$\lim_{x \to c} g (x) = L and \lim_{y \to L} f (y) = M$

both exist, with the stated values (The $ε$ - $δ$ limit $\lim_{x \to c} f (x) = L$ of $f : A \to R$ at a limit point $c$ of $A$ ). Suppose in addition that at least one of the following holds:

(i) $L \in B$ and $f (L) = M$ ;
(ii) there is a real $η > 0$ with $g (x) \neq L$ for every $x \in A$ satisfying $0 < ∣ x - c ∣ < η$ .

Then the limit of $f \circ g$ at $c$ exists, and

$\lim_{x \to c} f (g (x)) = \lim_{y \to L} f (y) = M .$

At least one extra hypothesis is necessary. With both omitted the statement is false, and FALSE: $\lim_{x \to c} f (g (x)) = M$ whenever $\lim_{x \to c} g = L$ and $\lim_{y \to L} f = M$ refutes it with a two-line witness in which (i) fails because $f (L) \neq M$ and (ii) fails because $g$ is constantly equal to $L$ .

Why an extra hypothesis is needed at all. The inner limit controls $g (x)$ only up to $∣ g (x) - L ∣ < ρ$ ; it does not prevent $g (x)$ from equalling $L$ . But The $ε$ - $δ$ limit $\lim_{x \to c} f (x) = L$ of $f : A \to R$ at a limit point $c$ of $A$ says nothing about $f$ at the point $L$ , so the outer estimate is unavailable exactly at the values $g (x) = L$ . Hypothesis (i) supplies the missing value directly; hypothesis (ii) excludes those values.

Facts & Assumptions

Given: Sets $A, B \subseteq R$ , functions $g : A \to R$ with $g (A) \subseteq B$ and $f : B \to R$ , a limit point $c$ of $A$ , a limit point $L$ of $B$ , and reals with $\lim_{x \to c} g (x) = L$ and $\lim_{y \to L} f (y) = M$ ; and the assumption that (i) or (ii) of the statement holds (The $ε$ - $δ$ limit $\lim_{x \to c} f (x) = L$ of $f : A \to R$ at a limit point $c$ of $A$ , Limit point, isolated point, adherent point, derived set, and dense subset of $R$ ).

[L1]

The limit condition: $\lim_{x \to c} h (x) = P$ 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 $∣ h (x) - P ∣ < ε$ (The $ε$ - $δ$ limit $\lim_{x \to c} f (x) = L$ of $f : A \to R$ at a limit point $c$ of $A$ ).

[L2]

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

[L3]

Order arithmetic: of two positive reals the smaller is positive, the order being total; and trichotomy (Ordered field).

[L4]

Limit point, and neighbourhoods (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 $R$ ).

Proof

technique · cases

1.1

Let $ε > 0$ be an arbitrary real. By [L1] applied to $f$ at $L$ , fix a real $ρ > 0$ such that every $y \in B$ with $0 < ∣ y - L ∣ < ρ$ satisfies $∣ f (y) - M ∣ < ε$ ; then by [L1] applied to $g$ at $c$ , with $ρ$ in the role of the tolerance, fix a real $δ_{1} > 0$ such that every $x \in A$ with $0 < ∣ x - c ∣ < δ_{1}$ satisfies $∣ g (x) - L ∣ < ρ$ .

L1choose

2.1

Case (i): assume $L \in B$ and $f (L) = M$ , and put $δ : = δ_{1} > 0$ . Let $x \in A$ with $0 < ∣ x - c ∣ < δ$ and set $y : = g (x)$ , an element of $B$ since $g (A) \subseteq B$ ; then $∣ y - L ∣ < ρ$ . If $y = L$ then $∣ f (y) - M ∣ = ∣ f (L) - M ∣ = ∣ 0 ∣ = 0 < ε$ ; and if $y \neq L$ then $0 < ∣ y - L ∣ < ρ$ , so $∣ f (y) - M ∣ < ε$ . In both events $∣ (f \circ g) (x) - M ∣ < ε$ .

step 1.1assume-case valueL1L2L3

2.2

Case (ii): assume there is a real $η > 0$ with $g (x) \neq L$ for every $x \in A$ satisfying $0 < ∣ x - c ∣ < η$ , and let $δ$ be the smaller of $δ_{1}$ and $η$ , so $δ > 0$ . Let $x \in A$ with $0 < ∣ x - c ∣ < δ$ and set $y : = g (x) \in B$ ; then $y \neq L$ , so $∣ y - L ∣ > 0$ , and $∣ y - L ∣ < ρ$ , so $0 < ∣ y - L ∣ < ρ$ and $∣ (f \circ g) (x) - M ∣ = ∣ f (y) - M ∣ < ε$ .

step 1.1assume-case avoidL1L2L3

3.1

By hypothesis at least one of (i) and (ii) holds, so in either case a real $δ > 0$ has been produced with $∣ (f \circ g) (x) - M ∣ < ε$ for every $x \in A$ satisfying $0 < ∣ x - c ∣ < δ$ ; since $ε > 0$ was arbitrary and $c$ is a limit point of $A$ , the limit of $f \circ g$ at $c$ exists and equals $M$ .

step 2.1step 2.2L1L4cases-exhaustive∎

Remarks

The hypothesis that $L$ is a limit point of $B$ is what makes $\lim_{y \to L} f (y)$ meaningful at all (The $ε$ - $δ$ limit $\lim_{x \to c} f (x) = L$ of $f : A \to R$ at a limit point $c$ of $A$ ); it is not an extra assumption of convenience. Note that it does not follow from $\lim_{x \to c} g (x) = L$ : a constant $g$ has that limit while $B$ may be a set for which $L$ is isolated.
Hypothesis (i) is the continuity hypothesis in disguise. Saying $L \in B$ and $f (L) = M = \lim_{y \to L} f (y)$ is exactly saying that $f$ is continuous at $L$ in the sense the next page of this track will define; that is the form in which this theorem is usually quoted, and it is why textbook statements of "the limit of a composition" almost always assume continuity of the outer function.
Hypothesis (ii) is the one that survives without continuity, and it is the hypothesis under which substitutions such as $y = 1 / x$ are legitimate: there the inner function omits the critical value on a punctured neighbourhood for a structural reason, not by assumption on $f$ .
The two hypotheses are genuinely different, neither implying the other. The companion page exhibits a pair satisfying neither, and the same pair with the inner function replaced by the identity, which satisfies (ii) but not (i).

Depends on

Used by

Dependency tree · two levels

16 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)
J. Lebl, Basic Analysis I, §3.1 (standard reference, not scraped)
T. Tao, Analysis I, 3rd ed., §9.3 (standard reference, not scraped)

Composition of limits holds under either hypothesis: ff is defined at LL with value MM, or gg avoids LL on a punctured neighbourhood of cc