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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 .
In mathematics, the notions of an absolutely monotonic function and a completely monotonic function are two very closely related concepts. Both imply very strong monotonicity properties. Both imply very strong monotonicity properties.
In statistics and numerical analysis, isotonic regression or monotonic regression is the technique of fitting a free-form line to a sequence of observations such that the fitted line is non-decreasing (or non-increasing) everywhere, and lies as close to the observations as possible.
A perfectly monotonic increasing relationship implies that for any two pairs of data values X i, Y i and X j, Y j, that X i − X j and Y i − Y j always have the same sign. A perfectly monotonic decreasing relationship implies that these differences always have opposite signs. The Spearman correlation coefficient is often described as being ...
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 .
Examples of monotone submodular functions include: Linear (Modular) functions Any function of the form () = is called a linear function. Additionally if , then f is monotone. Budget-additive functions
In mathematics, a homothetic function is a monotonic transformation of a function which is homogeneous; [2] however, since ordinal utility functions are only defined up to an increasing monotonic transformation, there is a small distinction between the two concepts in consumer theory. [1]: 147
A sequence of functions () converges uniformly to when for arbitrary small there is an index such that the graph of is in the -tube around f whenever . The limit of a sequence of continuous functions does not have to be continuous: the sequence of functions () = (marked in green and blue) converges pointwise over the entire domain, but the limit function is discontinuous (marked in red).