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