A monotonic function is a function which is either entirely nonincreasing or nondecreasing.
A real-valued function that is continuous
on an interval and differentiable
in its interior is nondecreasing if 
throughout the interior and
nonincreasing if 
throughout the interior. These conditions are also
necessary for their respective kinds of monotonicity, by the definition of the derivative. The mean-value
theorem proves sufficiency without requiring the derivative
to be continuous.
A strictly monotonic function is either strictly increasing or strictly
decreasing. Thus, for any 
in its domain, either 
always holds or 
always holds. A positive derivative
throughout the interior of an interval guarantees strict
increase. The strictly increasing function 
has 
, so strict increase can also occur when the
derivative vanishes at an isolated point.
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, p. 144). 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 elements of the domain to non-decreasing values
of the codomain. In particular, if 
is a set function
from a set 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.
