Monotonic function
In mathematics, a function (mathematics) is monotonic or monotone increasing if it preserves order: that is, if inputs x and y satisfy Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x \le y} then the outputs from f satisfy Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f(x) \le f(y)} . A monotonic decreasing function similarly reverses the order. A function is strictly monotonic if inputs x and y satisfying Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x < y} have outputs from f satisfying Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f(x) < f(y)} : that is, it is injective in addition to being montonic.
A differentiable function on the real numbers is monotonic when its derivative is non-zero: this is a consequence of the Mean Value Theorem.
Monotonic sequence
A special case of a monotonic function is a sequence regarded as a function defined on the natural numbers. So a sequence Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle a_n} is monotonic increasing if Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle m \le n} implies Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle a_m \le a_n} . In the case of real sequences, a monotonic sequence converges if it is bounded. Every real sequence has a monotonic subsequence.
References
- A.G. Howson (1972). A handbook of terms used in algebra and analysis. Cambridge University Press, 115,119. ISBN 0-521-09695-2.