CounterexampleConstruction: AI-adaptedVerification: AI-generatedSession-authored (Fable 5 assisted)precheck passaudited 2026-08-02

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

[L2]

$\log(xy)=\log x+\log y$ and $\log2>0$ (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 $\delta>0$ , some natural $n\ge1$ satisfies $1/n<\delta$ (For every $\varepsilon > 0$ in a complete ordered field there is a natural $n \ge 1$ with $1/n < \varepsilon$ ).

Counterexample

technique · direct

1.1

For $n\ge1$ , put $x_n=1/n$ and $y_n=2/n$ . Then $|x_n-y_n|=1/n$ .

givenalgebra

2.1

The logarithm gap is $|\log y_n-\log x_n|=|\log2|=\log2$ .

step 1.1L2

3.1

With $\varepsilon=\log2/2$ , every $\delta>0$ admits an $n$ from [L3] for which the pair in step 1.1 is within $\delta$ but its image gap exceeds $\varepsilon$ .

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 · next 3 levels

Direct dependencies and their dependencies through the next three levels: 52 results over 16 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, Logarithm and Exponential (standard reference, not scraped)
MIT OpenCourseWare 18.100B Real Analysis, Spring 2025 full lecture notes (standard reference, not scraped)