Where are functions increasing or decreasing?
In the following, suppose y = f(x) is real-valued function of a real variable x whose domain includes a set S. • Definition: f is constant on S if and only if for each pair of real numbers a and b in S, f(a) = f(b) Example of a function constant on an interval [c,d] • Definition: f is increasing on S if and only if for each pair of real numbers a and b in S, a < b implies f(a) < f(b) Example of a function strictly increasing on interval [c,d] • Definition: f is strictly increasing on S if and only if for each pair of real numbers a and b in S, a < b implies f(a) < f(b) Example of a function increasing on interval [c,d] but not strictly increasing as it is constant on two subintervals • Definition: f is strictly decreasing on S if and only if for each pair of real numbers a and b in S, a < b implies f(a) > f(b) Example of a function strictly decreasing on interval [c,d] • Definition: f is decreasing on S if and only if for each pair of real numbers a and b in S, a < b implies f(a) > f(b
