The following properties are true for a monotonic function f : R → R:
* f has limits from the right and from the left at every point of its domain;
* f has a limit at infinity (either ∞ or −∞) of either a real number, ∞, or −∞.
* f can only have jump discontinuities;
* f can only have countably many discontinuities in its domain.
These properties are the reason why monotonic functions are useful in technical work in analysis. Two facts about these functions are:
* if f is a monotonic function defined on an interval I, then f is differentiable almost everywhere on I, i.e. the set of numbers x in I such that f is not differentiable in x has Lebesgue measure zero.
* if f is a monotonic function defined on an interval [a, b], then f is Riemann integrable.
An important application of monotonic functions is in probability theory. If X is a random variable, its cumulative distribution function
FX(x) = Prob(X ≤ x)
is a monotonically increasing function.
A function is unimodal if it is monotonically increasing up to some point (the mode) and then monotonically decreasing.