# Definition:Strictly Monotone

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## Definition

### Ordered Sets

Let $\left({S, \preceq_1}\right)$ and $\left({T, \preceq_2}\right)$ be ordered sets.

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