Monotonic and Strictly Monotonic Functions

A function
$f(x)$
is monotonic (increasing) if
$\frac{d(f(x))}{dx} \ge 0$
or monotonic (decreasing) if
$\frac{d(f(x))}{dx} \le 0$
on its domain.
It is strictly monotonic (increasing) if
$\frac{d(f(x))}{dx} \gt 0$
and strictly monotonic (decreasing) if
$\frac{d(f(x))}{dx} \lt 0$
on its domain.
It is not enough that
$\frac{d(f(x))}{dx} \neq 0$
for a function to be strictly monotonic.
$f(x)= \frac{1}{x}$
is strictly monotonic (decreasing) for
$x \lt 0$
and strictly monotonic (increasing) for
$x \gt 0$
but at
$x=0$

$f(x)$
is not defined.
A monotonic function has no turning points, though it can have stationary points of inflection (
$\frac{d(f(x))}{dx}= \frac{d^2 (f(x))}{dx^2}=0$
). If it is strictly monotonic, it has no stationary points so that
$\frac{d(f(x))}{dx} \neq 0$
.