Definition: We say that a $f$ is
(montonically) increasing for a set D, if whenever
$a < b $ for $a$ and $ b \in D$ , $f(a) < f(b)$.
We say that a $f$ is (montonically) decreasing
if whenever $a < b $ for $a$ and $ b \in D$, $f(a) > f(b)$. Definition: We say that a $f$ is monotonic for a set D, if either $f$ is (monotically) increasing or $f$ is (monotonically) decreasing
for the set $D$.
Graph
Mapping Diagram
Increasing:
If $a < b $ then $f(a) < f(b)$.
Decreasing:
If $a < b $ then $f(a) > f(b)$.
Definition: We say that a $f$ is monotonic at a number or point $c$, if either $f$ is (monotically) increasing or $f$ is (monotonically) decreasing
for an open interval $D = (a,b)$ with $c \in D$.
An arrow on the mapping diagram target axis at the point representing the value of the function at $c$ can be used to indicate
whether $f$ is increasing (pointing up) or
decreasing (pointing down) at $c$.
When appropriate the magnitude of such an arrow can indicate the rate of increase or decrease of the function values.