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{{DISPLAYTITLE:Monotonic Function}}

A '''monotonic function''' is a function that preserves the given order. In the context of real-valued functions of a real variable, a function is called monotonic if it is either entirely non-increasing or non-decreasing. Monotonic functions are important in various fields such as mathematics, economics, and computer science due to their properties and applications.

==Definition==
A function \( f: \mathbb{R} \to \mathbb{R} \) is said to be:

* '''Monotonically increasing''' if for all \( x \leq y \), \( f(x) \leq f(y) \).
* '''Monotonically decreasing''' if for all \( x \leq y \), \( f(x) \geq f(y) \).

If the inequality is strict (i.e., \( f(x) < f(y) \) or \( f(x) > f(y) \)), the function is said to be '''strictly monotonic'''.

==Properties==
Monotonic functions have several important properties:

* '''Continuity''': While monotonic functions can be discontinuous, they can only have jump discontinuities. They cannot oscillate between values.
* '''Limits''': If a function is monotonic on an interval, it has limits at the endpoints of the interval.
* '''Invertibility''': A strictly monotonic function is invertible on its domain.

==Examples==

===Monotonically Increasing Functions===
* The function \( f(x) = x^2 \) is monotonically increasing on the interval \( [0, \infty) \).
* The exponential function \( f(x) = e^x \) is monotonically increasing on \( \mathbb{R} \).

[[File:Monotonicity_example1.svg|Example of a monotonically increasing function|thumb|right]]

===Monotonically Decreasing Functions===
* The function \( f(x) = -x \) is monotonically decreasing on \( \mathbb{R} \).
* The function \( f(x) = \frac{1}{x} \) is monotonically decreasing on the interval \( (0, \infty) \).

[[File:Monotonicity_example2.svg|Example of a monotonically decreasing function|thumb|left]]

===Non-Monotonic Functions===
* The function \( f(x) = \sin(x) \) is not monotonic on \( \mathbb{R} \) because it oscillates between -1 and 1.

==Applications==
Monotonic functions are used in various applications:

* '''Economics''': In economics, utility functions are often assumed to be monotonic, reflecting the idea that more of a good is better.
* '''Computer Science''': Monotonic functions are used in algorithms and data structures, such as priority queues and monotonic stack algorithms.
* '''Mathematics''': Monotonicity is a key concept in calculus and analysis, particularly in the study of limits and integrals.

==Monotonic Sequences==
A sequence \( \{a_n\} \) is called monotonic if it is either non-increasing or non-decreasing. Monotonic sequences have properties similar to monotonic functions, such as convergence properties.

[[File:Monotonic_dense_jumps_svg.svg|Example of a function with dense jumps|thumb|right]]

==Related Concepts==

===Monotonicity in Order Theory===
In order theory, a function between two ordered sets is monotonic if it preserves the order. This concept is more general than monotonic functions of real variables.

[[File:Hasse3_x_impl_y_and_z.svg|Hasse diagram illustrating monotonicity in order theory|thumb|left]]

===Growth Functions===
Growth functions describe how a quantity increases or decreases over time. Monotonic growth functions are particularly important in modeling and analysis.

[[File:Growth_equations.png|Equations representing growth functions|thumb|right]]

==Related Pages==
* [[Monotonicity criterion]]
* [[Monotonic transformation]]
* [[Order theory]]
* [[Sequence (mathematics)]]

[[Category:Mathematical analysis]]
[[Category:Functions and mappings]]

Monotonic function - Revision history

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