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In the case of finitely many jump discontinuities, f is a step function. The examples above are generalised step functions; they are very special cases of what are called jump functions or saltus-functions. [8] [9] More generally, the analysis of monotone functions has been studied by many mathematicians, starting from Abel, Jordan and Darboux.
The set of all points of discontinuity of a function may be a discrete set, a dense set, or even the entire domain of the function. The oscillation of a function at a point quantifies these discontinuities as follows: in a removable discontinuity, the distance that the value of the function is off by is the oscillation; in a jump discontinuity ...
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.
hide. In mathematics, Helly's selection theorem (also called the Helly selection principle) states that a uniformly bounded sequence of monotone real functions admits a convergent subsequence. In other words, it is a sequential compactness theorem for the space of uniformly bounded monotone functions. It is named for the Austrian mathematician ...
In mathematics, the Dirichlet–Jordan test gives sufficient conditions for a real -valued, periodic function f to be equal to the sum of its Fourier series at a point of continuity. Moreover, the behavior of the Fourier series at points of discontinuity is determined as well (it is the midpoint of the values of the discontinuity).
Dirichlet function. In mathematics, the Dirichlet function[1][2] is the indicator function of the set of rational numbers , i.e. if x is a rational number and if x is not a rational number (i.e. is an irrational number). It is named after the mathematician Peter Gustav Lejeune Dirichlet. [3] It is an example of a pathological function which ...
A monotone jump (or saltus) function is one with countably many jump discontinuities, enumerated as . At points of continuity ; otherwise the jumps are given by and . In this case can be written in the explicit form. Given an arbitrary monotone function with discontinuities at ( ), its jump data is given by and .
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 of real numbers converges to ...