Real Function Differentiability1

The terminology and notation used here have been introduced in the following articles: [11], [2], [8], [3], [4], [1], [5], [6], [7], [10], and [9]. For simplicity we follow the rules: x, x0, r, p will be real numbers, n will be a natural number, Y will be a subset of , Z will be a real open subset, X will be a set, s1 will be a sequence of real numbers, and f , f1, f2 will be partial functions from to . We now state the proposition (1) For every r holds r ∈ Y if and only if r ∈ if and only if Y = . A sequence of real numbers is called a real sequence convergent to 0 if: it is non-zero and it is convergent and lim it = 0. The following proposition is true (2) For every s1 holds s1 is a real sequence convergent to 0 if and only if s1 is non-zero and s1 is convergent and lim s1 = 0. A sequence of real numbers is called a constant real sequence if: it is constant. We now state the proposition (3) For every s1 holds s1 is a constant real sequence if and only if s1 is constant. In the sequel h will be a real sequence convergent to 0 and c will be a constant real sequence. A partial function from to is called a rest if: