Revised and upgraded Continuity 2 in HD. The real number line as a unification of geometry and analysis: The coordinate function as a given continuous function between a Euclidean line and the real numbers. Construction of continuous functions between geometric lines and line segments; continuous mappings between lines and circles; construction of real valued functions points on geometric objects, construction of real valued functions of real variables. We make use of the plane geometry of Euclid and the Dedekind concept of the real number line to construct continuous functions between geometric object, geometric objects and the real numbers, parameterize plane curves, and real valued continuous functions of a real variable.
