Continuity
continuity
Term used in a general sense by theologians to designate the successive existence of the Church from its foundation by Jesus Christ to the present day without a change in its dogma or hierarchical constitution. Specifically, it was a theory advanced by the High-Church Anglicans in the 19th century when they claimed communion with the Holy See on the grounds that their Church was identical with the pre-Reformation Church in England , hence with the Catholic Church. The absurdity of this claim is only too apparent, and never was it exposed more forcibly than by Cardinal Bourne at the thirteenth centenary of York.
Fuente: New Catholic Dictionary
Continuity
A class is said to be compactly (or densely) ordered by a relation R if it is ordered by R (see Order) and, whenever xRz and x?z, there is a y, not the same as either x or z, such that xRy and yRz. (Compact order may thus be described by saying that between any two distinct members of the class there is always a third, or by saying that no member has a next following member in the order.)
If a class b ii ordered by a relation R, and a ? b, we say that z is an upper bound of a if, for all x, x?a implies xRz; and that z is a least upper bound of a if z is an upper bound of a and there is no upper bound y of a, different from z, such that yRz.
A class b ordered by a relation R is said to have continuous order (Dedekindian continuity) if it is compactly ordered by R and every non-empty class a, for which a ? b, and which has an upper bound, has a least upper bound.
An important mathematical example of continuous order is afforded by the real numbers, ordered by the relation not greater than. According to usual geometric postulates, the points on a straight line also have continuous order, and, indeed, have the same order type as the real numbers.
The term continuity is also employed in mathematics in connection with functions of various kinds. We shall state the definition for the case of a monadic function f for which the range of the independent variable and the range of the dependent variable both consist of real numbers (see the article Function).
Let us use R for the relation not greater than among real numbers. A neighborhood of a real number c is determined by two real numbers m and n — both different from c and such that mRc and cRn — and is the class of real numbers x, other than m and n, such that mRx and xRn. The function f is said to be continuous at the real number c if the three following conditions are satisfied
c belongs to the range of the independent variable;
in every neighborhood of c there are numbers other than c belonging to the range of the independent variable;
corresponding to every neighborhood b of f(c) there is a neighborhood a of c such that, for every real number x belonging to the range of the independent variable, x?a implies f(x) ? b.
A function may be called continuous if it is continuous at every real number, or at every real number in a certain set determined by the context. — A.C.
E. V. Huntington, The Continuum, Cambridge, Mass., 1917.