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.