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:
“While it may not heighten our sympathy, wit widens our horizons by its flashes, revealing remote hidden affiliations and drawing laughter from far afield; humor, in contrast, strikes up fellow feeling, and though it does not leap so much across time and space, enriches our insight into the universal in familiar things, lending it a local habitation and a name.”
—Marie Collins Swabey. Comic Laughter, ch. 5, Yale University Press (1961)
“Every society consists of men in the process of developing from children into parents. To assure continuity of tradition, society must early prepare for parenthood in its children; and it must take care of the unavoidable remnants of infantility in its adults. This is a large order, especially since a society needs many beings who can follow, a few who can lead, and some who can do both, alternately or in different areas of life.”
—Erik H. Erikson (1904–1994)
“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)
“Truly man is a marvelously vain, diverse, and undulating object. It is hard to found any constant and uniform judgment on him.”
—Michel de Montaigne (1533–1592)