The problem from the proof outline from 11/28 covered the definition of a nondecreasing function. We learned about the nondecreasing sequences but not nondecreasing functions in this course. In this post, I would like to review the definition of a non-decreasing function and further explore relative definitions of a nonincreasing function, an increasing function, a decreasing function.
Nondecreasing function: A function f(x) is said to be nondecreasing on an interval I if for all b>a, where
Nonincreasing function: a function f(x) is said to be nonincreasing on an interval I if for all b>a with .
Increasing function:A function f(x) increases on an interval I if for all b>a, where . If , f(b)>f(a), for all b>a, the function is said to be strictly increasing.
Decreasing function: a function f(x) decreases on an interval I if for all b>a with . If f(b)<f(a), for all b>a, the function is said to be strictly decreasing.
If the derivative f'(x) of a continuous function f(x) satisfies ,f'(x)>0 on an open interval (a,b), then f(x) is increasing on (a,b). However, a function may increase on an interval without having a derivative defined at all points. For example, the function (see the graph below)is increasing everywhere, including the origin x=0, despite the fact that the derivative is not defined at that point.
This graph also demonstrates the features of nondecreasing functions, nonincreasing functions, increasing functions and decreasing functions.
Jeffreys, H. and Jeffreys, B. S. “Increasing and Decreasing Functions.” §1.065 in Methods of Mathematical Physics, 3rd ed. Cambridge, England: Cambridge University Press, p. 22, 1988.