Uniform Continuity - Local Continuity Versus Global Uniform Continuity

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):