# Monotonic function

A function that either never decreases or never increases. A non-decreasing, or isotonic, function satisfies: $LaTeX: f(x') \ge f(x)$ whenever $LaTeX: x' \ge x$ (it is strictly increasing if $LaTeX: f(x') > f(x)$ for $LaTeX: x' > x$). A non-increasing, or anatonic, function satisfies: $LaTeX: f(x') \le f(x)$ whenever $LaTeX: x' \ge x$ (it is strictly decreasing if $LaTeX: f(x') < f(x)$ for $LaTeX: x'> x$). This extends to a vector function, where $LaTeX: \mbox{range}(f)$ is in $LaTeX: \mathbb{R}^n$: $LaTeX: (f(x)-f(x))^T (x-y) \ge 0.$