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.

Further thoughts:

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.

Reference:

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.