CONTINUITY. Let f (x) be a function of x defined in a domain D, where D is either an interval for a real variable x or a region for a complex variable x; and let x=a be a point of D so that f (a) is a defined value of f (x). Then f (x) is said to be continuous at the point a if it is true that, for every positive number E however small, there exists a positive number de pendent on E such that f(a+n) —f (a) is numerically less than E for all values of n numerically less than 5, and such that al-0 belongs to D. By employing the notion of limit (see Limrr) this definition may be stated more briefly as follows: If x=a is a point of the domain D of definition of f(x) so that f (a) is a defined value of f (x) and if the limit of f(x), as x approaches a in D, exists and is f (a), then f (x) is said to be continuous at a.
If f(x) is continuous at each point of D then f(x) is said to be continuous throughout D. The number 45 which appears in the first form of the definition may change with a change of a as well as with a change of 77. If f (x) is such that can be taken independent of a and dependent only on 77, then f(x) is said to be uniformly continuous in D. A function which is continuous in a closed domain D is also uniformly continuous in D; but this proposition does not hold for an open domain. A real-valued function which is continuous in a closed interval has a greatest value and a least value which are actually attained in the interval, and there is at least one point in the interval at which the function takes any given value between its greatest and its least value.
The term geometric continuity is used for a concept which arises in projective geometry in introducing elements at infinity (see GEOMETRY).
See E. W. Hobson, Theory of Functions of a Real Variable, vol. i. (third edition, 1927) and vol. ii. (second edition, 1926). (R. D. CA.)