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A function that is not monotonic. In mathematics, a monotonic function (or monotone function) is a function between ordered sets that preserves or reverses the given order. [ 1 ][ 2 ][ 3 ] This concept first arose in calculus, and was later generalized to the more abstract setting of order theory.
Monotone convergence theorem. In the mathematical field of real analysis, the monotone convergence theorem is any of a number of related theorems proving the good convergence behaviour of monotonic sequences, i.e. sequences that are non- increasing, or non- decreasing. In its simplest form, it says that a non-decreasing bounded -above sequence ...
"Monotone" (Japanese: モノトーン, Hepburn: Monotōn) is a song by Japanese duo Yoasobi for the 2024 Japanese animated film Fureru as a theme. It was released as a single on October 1, 2024, by Echoes and Sony Music Entertainment Japan .
Operator monotone function. In linear algebra, the operator monotone function is an important type of real-valued function, fully classified by Charles Löwner in 1934. [1] It is closely allied to the operator concave and operator concave functions, and is encountered in operator theory and in matrix theory, and led to the Löwner–Heinz ...
Discontinuities of monotone functions. In the mathematical field of analysis, a well-known theorem describes the set of discontinuities of a monotone real-valued function of a real variable; all discontinuities of such a (monotone) function are necessarily jump discontinuities and there are at most countably many of them.
Informally, a measure has the property of being monotone in the sense that if is a subset of , the measure of is less than or equal to the measure of . Furthermore, the measure of the empty set is required to be 0. A simple example is a volume (how big an object occupies a space) as a measure.
Monotone cubic interpolation. hide. In the mathematical field of numerical analysis, monotone cubic interpolation is a variant of cubic interpolation that preserves monotonicity of the data set being interpolated. Monotonicity is preserved by linear interpolation but not guaranteed by cubic interpolation.
Dini's theorem. In the mathematical field of analysis, Dini's theorem says that if a monotone sequence of continuous functions converges pointwise on a compact space and if the limit function is also continuous, then the convergence is uniform. [1]