Based on the order relation for real numbers, real functions \(f:\mathbb R\mapsto\mathbb R\) can be classified by the way the values of \(f(x)\) increase (decrease), depending on increasing \(x\). The function \(f\) is called:
monotonically increasing, if for all \(x\) and \(y\) such that \(x \le y\) one has \(f(x) \le f(y)\),
strictly monotonically increasing, if for all \(x\) and \(y\) such that \(x < y\) one has \(f(x) < f(y)\),
monotonically decreasing, if for all \(x\) and \(y\) such that \(x \le y\) one has \(f(x) \ge f(y)\),
strictly monotonically decreasing, if for all \(x\) and \(y\) such that \(x y\) one has \(f(x) > f(y)\).
If \(f\) is either (strictly) monotonically increasing or decreasing, it is called monotonic.
Definitions: 1
Lemmas: 2
Proofs: 3 4 5 6 7 8 9
Propositions: 10 11 12 13 14 15 16 17 18 19 20
Sections: 21