Real–Valued Functions

The terminology and notation used here have been introduced in the following articles: [9], [12], [1], [10], [11], [13], [14], [2], [3], [4], [6], [5], [8], and [7]. Let r be a real number. Observe that r r is non negative. Let r be a real number. Observe that r · r is non negative and r · r is non negative. Let r be a non negative real number. One can check that √ r is non negative. Let r be a positive real number. Observe that √ r is positive. We now state the proposition (1) For every function f and for every set A such that f is one-to-one and A ⊆ dom(f) holds f(f)A = A. Let f be a non-empty function. One can verify that f−1({0}) is empty. Let R be a binary relation. We say that R is positive yielding if and only if: (Def. 1) For every real number r such that r ∈ rng R holds 0 < r. We say that R is negative yielding if and only if: (Def. 2) For every real number r such that r ∈ rng R holds 0 > r. We say that R is non-positive yielding if and only if: (Def. 3) For every real number r such that r ∈ rng R holds 0 ≥ r. We say that R is non-negative yielding if and only if: (Def. 4) For every real number r such that r ∈ rng R holds 0 ≤ r. Let X be a set and let r be a positive real number. Observe that X 7−→ r is positive yielding. Let X be a set and let r be a negative real number. Note that X 7−→ r is negative yielding.