Editing Uniform continuity (section)

== Examples and nonexamples ==

=== Examples ===
* [[Linear function]]s <math>x \mapsto ax + b</math> are the simplest examples of uniformly continuous functions.
* Any continuous function on the interval <math>[0,1]</math> is also uniformly continuous, since <math>[0,1]</math> is a compact set.
* If a function is differentiable on an open interval and its derivative is bounded, then the function is uniformly continuous on that interval.
* Every [[Lipschitz continuous]] map between two metric spaces is uniformly continuous. More generally, every [[Hölder continuous]] function is uniformly continuous.
* The [[absolute value|absolute value function]] is uniformly continuous, despite not being differentiable at <math>x = 0</math>. This shows uniformly continuous functions are not always differentiable.
* Despite being nowhere differentiable, the [[Weierstrass function]] is uniformly continuous.
* Every member of a [[Uniform equicontinuity|uniformly equicontinuous]] set of functions is uniformly continuous.

=== Nonexamples ===
* Functions that are unbounded on a bounded domain are not uniformly continuous. The [[tangent function]] is continuous on the interval <math>(-\pi/2, \pi/2)</math> but is ''not'' uniformly continuous on that interval, as it goes to infinity as <math>x \to \pi/2</math>.
* Functions whose derivative tends to infinity as <math>x</math> grows large cannot be uniformly continuous. The exponential function <math>x \mapsto e^x</math> is continuous everywhere on the real line but is not uniformly continuous on the line, since its derivative is <math>e^x</math>, and <math>e^x\to\infty</math> as <math>x \to \infty</math>.