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  
\[\frac{d(f(x))}{dx} \neq 0\]
.