CounterexampleConstruction: Literature-sourcedVerification: AI-adaptedprecheck passverified 2026-08-10 (gpt-5.6-terra-codex-subscription)

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Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
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The exponential is not uniformly continuous on $R$

Statement refuted

The exponential function is uniformly continuous on $R$ .

Facts & Assumptions

Given: $\exp^{'} = \exp$ .

[L1]

Uniform continuity is Uniform continuity of $f : A \to R$ : one $δ$ serving every pair of points of $A$ .

[L2]

The exponential is strictly increasing (The exponential function is strictly increasing), the mean value theorem is The mean value theorem, as the case $g (x) = x$ of Cauchy's: for $f$ continuous on $[a, b]$ with $a < b$ and differentiable on $(a, b)$ there is $c \in (a, b)$ with $f (b) - f (a) = f^{'} (c) (b - a)$ , and exponential dominates polynomials (The exponential dominates every fixed nonnegative integer power at $+ \infty$ ).

[L3]

The reciprocal sequence $1 / ι (n)$ tends to $0$ when started at $n \geq 1$ (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 $n \geq 1$ , let $x_{n} = ι (n)$ and $y_{n} = ι (n) + 1 / ι (n)$ . Then $∣ y_{n} - x_{n} ∣ = 1 / ι (n) \to 0$ .

1.2

By the mean value theorem, $\exp (y_{n}) - \exp (x_{n}) = \exp (c_{n}) / ι (n)$ for some $c_{n} \in (x_{n}, y_{n})$ . Since exponential is increasing, this is at least $\exp (ι (n)) / ι (n)$ , which tends to $+ \infty$ by [L2].

L2given

2.1

Thus arbitrarily close pairs have image distances bounded away from $0$ , contradicting the uniform-continuity condition [L1].

step 1.1step 1.2L1∎

Remarks

On every compact interval, $\exp$ is uniformly continuous by Heine-Cantor in $R$ : a continuous real function on a compact subset of $R$ is uniformly continuous, proved $R$ -natively from sequential compactness; the failure is global.

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

41 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

MIT OpenCourseWare 18.100B Real Analysis, Spring 2025 full lecture notes (standard reference, not scraped)
J. Lebl, Basic Analysis, Analytic Functions (standard reference, not scraped)
UTSA Mathematics Research Wiki, Uniform Continuity (standard reference, not scraped)

The exponential is not uniformly continuous on RR