Examples
- Every Lipschitz continuous map between two metric spaces is uniformly continuous. In particular, every function which is differentiable and has bounded derivative is uniformly continuous. More generally, every Hölder continuous function is uniformly continuous.
- Every member of a uniformly equicontinuous set of functions is uniformly continuous.
- The tangent function is continuous on the interval (−π/2, π/2) but is not uniformly continuous on that interval.
- The exponential function x ex is continuous everywhere on the real line but is not uniformly continuous on the line.
Read more about this topic: Uniform Continuity
Famous quotes containing the word examples:
“Histories are more full of examples of the fidelity of dogs than of friends.”
—Alexander Pope (1688–1744)
“There are many examples of women that have excelled in learning, and even in war, but this is no reason we should bring ‘em all up to Latin and Greek or else military discipline, instead of needle-work and housewifry.”
—Bernard Mandeville (1670–1733)
“It is hardly to be believed how spiritual reflections when mixed with a little physics can hold people’s attention and give them a livelier idea of God than do the often ill-applied examples of his wrath.”
—G.C. (Georg Christoph)