Monotonic function

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Figure 2. A monotonically decreasing function
Figure 3. A function that is not monotonic

In mathematics, a monotonic function (or monotone function) is a function between ordered sets that preserves the given order. This concept first arose in calculus, and was later generalized to the more abstract setting of order theory.

Monotonicity in calculus and analysis[edit]

In calculus, a function f defined on a subset of the real numbers with real values is called monotonic if it is either entirely nonincreasing or nondecreasing. It is called monotonically increasing (also increasing or non-decreasing), if for all x and y such that x \leq y one has f\!\left(x\right) \leq f\!\left(y\right), so f preserves the order (see Figure 1). Likewise, a function is called monotonically decreasing (also decreasing, or non-increasing) if, whenever x \leq y, then f\!\left(x\right) \geq f\!\left(y\right), so it reverses the order (see Figure 2).

If the order \leq in the definition of monotonicity is replaced by the strict order <, then one obtains a stronger requirement. A function with this property is called strictly increasing. Again, by inverting the order symbol, one finds a corresponding concept called strictly decreasing. Functions that are strictly increasing or decreasing are one-to-one (because for x not equal to y, either x < y or x > y and so, by monotonicity, either f\!\left(x\right) < f\!\left(y\right) or f\!\left(x\right) > f\!\left(y\right), thus f\!\left(x\right) is not equal to f\!\left(y\right).)




A function f\!\left(x\right) is said to be absolutely monotonic over an interval \left(a, b\right) if the derivatives of all orders of f are nonnegative at all points on the interval.

Some basic applications and results[edit]

The following properties are true for a monotonic function f\colon \mathbb{R} \to \mathbb{R}:


An important application of monotonic functions is in probability theory. If X is a random variable, its cumulative distribution function F_X\!\left(x\right) = \text{Prob}\!\left(X \leq x\right) is a monotonically increasing function.


When f is a strictly monotonic function, then f is injective on its domain, and if T is the range of f, then there is an inverse function on T for f.

Monotonicity in functional analysis[edit]


(Tu - Tv, u - v) \geq 0 \quad \forall u,v \in X.



(w_1 - w_2, u_1 - u_2) \geq 0.


Monotonicity in order theory[edit]



xy implies f(x) ≥ f(y),




Monotonicity in the context of search algorithms[edit]


h(n) \leq c(n, a, n') + h(n').


Boolean functions[edit]

In Boolean algebra, a monotonic function is one such that for all ai and bi in {0,1}, if a1b1, a2b2, ..., anbn, then f(a1, ..., an) ≤ f(b1, ..., bn). In other words, a Boolean function is monotonic if, for every combination of inputs, switching one of the inputs from false to true can only cause the output to switch from false to true and not from true to false. Graphically, this means that a Boolean function is monotonic when in its Hasse diagram (dual of its Venn diagram), there is no 1 (red vertex) connected to a higher 0 (white vertex).



Monotonic logic[edit]

Monotonicity of entailment is a property of many logic systems that states that the hypotheses of any derived fact may be freely extended with additional assumptions. Any true statement in a logic with this property continues to be true, even after adding new axioms. Logics with this property may be called monotonic, to differentiate them from non-monotonic logic.

See also[edit]


  2. ^ Conditions for optimality: Admissibility and consistency pg. 94-95 (Russel & Norvig 2010).


  • Bartle, Robert G. (1976). The elements of real analysis (second edition ed.). 
  • Renardy, Michael and Rogers, Robert C. (2004). An introduction to partial differential equations. Texts in Applied Mathematics 13 (Second edition ed.). New York: Springer-Verlag. p. 356. ISBN 0-387-00444-0. 
  • Riesz, Frigyes and Béla Szőkefalvi-Nagy (1990). Functional Analysis. Courier Dover Publications. ISBN 978-0-486-66289-3. 
  • Simon, Carl P.; Blume, Lawrence (April 1994). Mathematics for Economists (first edition ed.). ISBN 978-0-393-95733-4.  (Definition 9.31)

External links[edit]