A monotonic function is a function which is either entirely nonincreasing or nondecreasing. A differentiable real-valued function on an interval is monotonic if its first derivative (which need not be continuous) does not change sign.
A strictly monotonic function is either strictly increasing or strictly
decreasing. Thus, for any 
in its domain, either 
always holds or 
always holds.
An absolutely monotonic function has nonnegative derivatives of every order on its interval of definition, including order zero, so the function itself must also be nonnegative (Widder 1941). Absolute monotonicity therefore implies that the function is nondecreasing, but does not require strict monotonicity. A positive constant function is an example.
The term monotonic may also be used to describe set functions which map subsets of the domain
to non-decreasing values of the codomain. In particular,
if 
is a set function from a collection
of sets 
to an ordered set 
, then 
is said to be monotone if whenever
as elements of 
, 
. This particular definition comes up frequently
in measure theory where many of the families of
functions defined (including outer
measure, premeasure, and measure)
begin by considering monotonic set functions.
