Definition:Strictly Monotone

Ordered Sets
Let $\left({S, \preceq_1}\right)$ and $\left({T, \preceq_2}\right)$ be posets.

Let $\phi: \left({S, \preceq_1}\right) \to \left({T, \preceq_2}\right)$ be a mapping.

Then $\phi$ is strictly monotone iff it is either strictly increasing or strictly decreasing.

Note that this definition also holds if $S = T$.

Real Functions
This definition continues to hold when $S = T = \R$.

Thus, let $f$ be a real function.

Then $f$ is strictly monotone iff it is either strictly increasing or strictly decreasing.

Sequences
Let $\left \langle {x_n} \right \rangle$ be a sequence in $\R$.

Then $\left \langle {x_n} \right \rangle$ is strictly monotone if it is either strictly increasing or strictly decreasing.

Also see

 * Increasing
 * Decreasing
 * Monotone