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 $f(x_{1})\leq f(x_{2})$.

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