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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.
cleanup Please help improve this article if you can. 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 ...
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.
Quantile functions are used in both statistical applications and Monte Carlo methods. The quantile function is one way of prescribing a probability distribution, and it is an alternative to the probability density function (pdf) or probability mass function, the cumulative distribution function (cdf) and the characteristic function.
Gradients of convex functions are cyclically monotone. In fact, the converse is true. [ 4 ] Suppose U {\displaystyle U} is convex and f : U ⇉ R n {\displaystyle f:U\rightrightarrows \mathbb {R} ^{n}} is a correspondence with nonempty values.
e. In mathematics, the integral of a non-negative function of a single variable can be regarded, in the simplest case, as the area between the graph of that function and the X axis. The Lebesgue integral, named after French mathematician Henri Lebesgue, is one way to make this concept rigorous and to extend it to more general functions.
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]
In this context, a property is monotone if it remains true when edges are added; for example, planarity is not monotone, but non-planarity is monotone. A stronger version of this conjecture, called the evasiveness conjecture or the Aanderaa–Karp–Rosenberg conjecture, states that exactly ( n 2 ) = n ( n − 1 ) / 2 {\displaystyle {\tbinom {n ...