Local Continuity Versus Global Uniform Continuity
Continuity itself is a local (more precisely, pointwise) property of a function—that is, a function f is continuous, or not, at a particular point. When we speak of a function being continuous on an interval, we mean only that it is continuous at each point of the interval. In contrast, uniform continuity is a global property of f, in the sense that the standard definition refers to pairs of points rather than individual points. On the other hand, it is possible to give a local definition in terms of the natural extension f*, see below.
The mathematical statements that a function is continuous on an interval I and the definition that a function is uniformly continuous on the same interval are structurally very similar. Continuity of a function for every point x of an interval can thus be expressed by a formula starting with the quantification
which is equivalent to
whereas for uniform continuity, the order of the second and third quantifiers is reversed:
(the domains of the variables have been deliberately left out so as to emphasize quantifier order). Thus for continuity at each point, one takes an arbitrary point x, and then there must exist a distance δ,
while for uniform continuity a single δ must work uniformly for all points x (and y):
Read more about this topic: Uniform Continuity
Famous quotes containing the words local, continuity, global and/or uniform:
“Savages cling to a local god of one tribe or town. The broad ethics of Jesus were quickly narrowed to village theologies, which preach an election or favoritism.”
—Ralph Waldo Emerson (1803–1882)
“Only the family, society’s smallest unit, can change and yet maintain enough continuity to rear children who will not be “strangers in a strange land,” who will be rooted firmly enough to grow and adapt.”
—Salvador Minuchin (20th century)
“The Sage of Toronto ... spent several decades marveling at the numerous freedoms created by a “global village” instantly and effortlessly accessible to all. Villages, unlike towns, have always been ruled by conformism, isolation, petty surveillance, boredom and repetitive malicious gossip about the same families. Which is a precise enough description of the global spectacle’s present vulgarity.”
—Guy Debord (b. 1931)
“I’ve always been impressed by the different paths babies take in their physical development on the way to walking. It’s rare to see a behavior that starts out with such wide natural variation, yet becomes so uniform after only a few months.”
—Lawrence Kutner (20th century)