Definition:Strictly Monotone

Definition

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 $\left({S, \preceq}\right)$ be a totally ordered set.

Then a sequence $\left \langle {a_k} \right \rangle_{k \in A}$ of terms of $S$ is strictly monotone if it is either strictly increasing or strictly decreasing.

Real Sequences

The above definition for sequences is usually applied to real number 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.

Notes

This can also be called strictly monotonic.

Also see

• Increasing
• Decreasing
• Monotone