CounterexampleConstruction: AI-adaptedVerification: AI-generatedprecheck passaudited 2026-08-02

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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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The logarithm is not uniformly continuous on the positive half-line

Statement refuted

The natural logarithm is uniformly continuous on $(0, \infty)$ .

Facts & Assumptions

Given: The positive half-line and the natural logarithm.

[L1]

Uniform continuity has one $δ > 0$ for all pairs in the domain (Uniform continuity of $f : A \to R$ : one $δ$ serving every pair of points of $A$ ).

[L2]

$\log (x y) = \log x + \log y$ and $\log 2 > 0$ (Order, continuity, range, and the product, quotient, and reciprocal laws for the natural logarithm, The natural logarithm as the inverse of the exponential function).

[L3]

For every $δ > 0$ , some natural $n \geq 1$ satisfies $1 / n < δ$ (For every $ε > 0$ in a complete ordered field there is a natural $n \geq 1$ with $1 / n < ε$ ).

Counterexample

technique · direct

1.1

For $n \geq 1$ , put $x_{n} = 1 / n$ and $y_{n} = 2 / n$ . Then $∣ x_{n} - y_{n} ∣ = 1 / n$ .

givenalgebra

2.1

The logarithm gap is $∣ \log y_{n} - \log x_{n} ∣ = ∣ \log 2 ∣ = \log 2$ .

step 1.1L2

3.1

With $ε = \log 2 / 2$ , every $δ > 0$ admits an $n$ from [L3] for which the pair in step 1.1 is within $δ$ but its image gap exceeds $ε$ .

step 1.1step 2.1L3

4.1

This contradicts [L1], so $\log$ is not uniformly continuous on $(0, \infty)$ .

step 3.1L1∎

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

18 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, Logarithm and Exponential (standard reference, not scraped)
MIT OpenCourseWare 18.100B Real Analysis, Spring 2025 full lecture notes (standard reference, not scraped)