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The exponential is not uniformly continuous on $\mathbb{R}$

Statement refuted

The exponential function is uniformly continuous on $\mathbb R$ .

Facts & Assumptions

Given: $\exp'=\exp$ .

[L1]

Uniform continuity is Uniform continuity of $f : A \to \mathbb{R}$ : one $\delta$ 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 $1/\iota(n)$ tends to $0$ when started at $n\ge1$ (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 $n\ge1$ , let $x_n=\iota(n)$ and $y_n=\iota(n)+1/\iota(n)$ . Then $|y_n-x_n|=1/\iota(n)\to0$ .

1.2

By the mean value theorem, $\exp(y_n)-\exp(x_n)=\exp(c_n)/\iota(n)$ for some $c_n\in(x_n,y_n)$ . Since exponential is increasing, this is at least $\exp(\iota(n))/\iota(n)$ , 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 $\mathbb{R}$ : a continuous real function on a compact subset of $\mathbb{R}$ is uniformly continuous, proved $\mathbb{R}$ -natively from sequential compactness; the failure is global.

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: 134 results over 21 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

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 R\mathbb{R}R