A function 
increases on an interval 
if 
for all 
, where 
. If 
for all 
, the function is said to be strictly
increasing.
Conversely, a function 
decreases on an interval 
if 
for all 
with 
. If 
for all 
, the function is said to be strictly
decreasing.
If the derivative 
of a continuous function 
satisfies 
on an open interval 
, then 
is increasing on 
. However, a function may increase on an interval without
having a derivative defined at all points. For example, the function 
is increasing everywhere, including the origin 
, despite the fact that the derivative
is not defined at that point.
