Definition:Strictly Monotone
Definition
Ordered Sets
Let $\struct {S, \preceq_1}$ and $\struct {T, \preceq_2}$ be ordered sets.
Let $\phi: \struct {S, \preceq_1} \to \struct {T, \preceq_2}$ be a mapping.
Then $\phi$ is strictly monotone if and only if 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 $S = T = \R$:
Let $f: S \to \R$ be a real function, where $S \subseteq \R$.
Then $f$ is strictly monotone if and only if it is either strictly increasing or strictly decreasing.
Sequences
Let $\struct {S, \preceq}$ be a totally ordered set.
Then a sequence $\sequence {a_k}_{k \mathop \in A}$ of elements of $S$ is strictly monotone if and only if it is either strictly increasing or strictly decreasing.
Notes
This can also be called strictly monotonic.