On the Continuity of Some Functions

For simplicity, we adopt the following rules: x, X are sets, i, n, m are natural numbers, r, s are real numbers, c, c1, c2, d are complex numbers, f , g are complex-valued functions, g1 is an n-element complex-valued finite sequence, f1 is an n-element real-valued finite sequence, T is a non empty topological space, and p is an element of En T. Let R be a binary relation and let X be an empty set. Observe that R◦X is empty and R−1(X) is empty. Let A be an empty set. Observe that every element of A is empty. We now state the proposition (1) For every trivial set X and for every set Y such that X ≈ Y holds Y is trivial. Let r be a real number. Observe that r2 is non negative. Let r be a positive real number. Note that r2 is positive. Let us note that √ 0 is zero.