Increasing function

Definition

An increasing function (sometimes called a strictly increasing function) is a function $f$ satisfying the following condition for any two distinct elements $x_1, x_2$ in its domain:

$x_1 < x_2 \implies f(x_1) < f(x_2)$

Note: Sometimes the term increasing is used for a non-decreasing function, where we do not require strict inequality but only insist on $f(x_1) \le f(x_2)$.

Also note that for functions whose domain is empty or a single point, the definition of being increasing is vacuously satisfied and is uninteresting.