Monotone Real Functions

Why

We can interpret a real function as tracing a path as we move from left to right in its domain.¹ We want language for whether this tracing increases or decreases the range values.

Definition

Let $A \subset \R $ and let $f: A \to \R $. A function is monotone increasing if $f(x) < f(y)$ whenever $x < y$, and monotone nondecreasing if $f(x) \leq f(y)$ whenever $x, y \in \R $ and $x < y$.² Similarly we define monotone decreasing and monotone nonincreasing.

1. Future editions will likely have this interpretation in a separate sheet. ↩︎
2. Unforunately, some authors use “monotone increasing for “monotone nondecreasing” and use the terminolgy strictly monotone increasing for “monotone increasing”.” ↩︎