Monotonicity
Theorem
A function \(f : A \to B\) between ordered sets is called monotonic increasing if it preserves inequalities:
\[ a \prec b \implies f(a) \prec f(b)\]
or monotonic decreasing if it reverses inequalities:
\[ a \prec b \implies f(a) \succ f(b).\]
Intuitively, a function is increasing if making the input bigger makes the output bigger, and similarly for a decreasing function.
In the specific case of a real valued function of a real number, there is a close relationship between monotonicity and the derivative.