More on the Continuity of Real Functions

The terminology and notation used here have been introduced in the following articles: [3], [7], [17], [2], [4], [12], [13], [14], [16], [1], [5], [9], [15], [18], [10], [8], [20], [21], [19], [11], [22], and [6]. For simplicity, we use the following convention: n, i denote elements of N, X, X1 denote sets, r, p, s, x0, x1, x2 denote real numbers, f , f1, f2 denote partial functions from R to Rn, and h denotes a partial function from R to the carrier of 〈En, ‖ · ‖〉. Let us consider n, f , x0. We say that f is continuous in x0 if and only if: (Def. 1) There exists a partial function g from R to the carrier of 〈En, ‖ · ‖〉 such that f = g and g is continuous in x0. We now state four propositions: (1) If h = f, then f is continuous in x0 iff h is continuous in x0. (2) If x0 ∈ X and f is continuous in x0, then f X is continuous in x0.