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The notions of completely and absolutely monotone function/sequence play an important role in several areas of mathematics. For example, in classical analysis they occur in the proof of the positivity of integrals involving Bessel functions or the positivity of Cesàro means of certain Jacobi series. [6]
For example, if = is strictly increasing on the range [,], then it has an inverse = on the range [(), ()]. The term monotonic is sometimes used in place of strictly monotonic , so a source may state that all monotonic functions are invertible when they really mean that all strictly monotonic functions are invertible.
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.
A function : defined on an interval is said to be operator monotone if whenever and are Hermitian matrices (of any size/dimensions) whose eigenvalues all belong to the domain of and whose difference is a positive semi-definite matrix, then necessarily () where () and () are the values of the matrix function induced by (which are matrices of the same size as and ).
Let be a real-valued monotone function defined on an interval. Then the set of discontinuities of the first kind is at most countable.. One can prove [5] [3] that all points of discontinuity of a monotone real-valued function defined on an interval are jump discontinuities and hence, by our definition, of the first kind.
Monotone refers to a sound, for example music or speech, that has a single unvaried tone. See pure tone and monotonic scale. Monotone or monotonicity may also refer to:
As a corollary, if is a ring of sets, then the smallest monotone class containing it coincides with the 𝜎-ring of .. By invoking this theorem, one can use monotone classes to help verify that a certain collection of subsets is a 𝜎-algebra.
Assume that L coincides with the x-axis.Then the leftmost and rightmost vertices of a monotone polygon decompose its boundary into two monotone polygonal chains such that when the vertices of any chain are being traversed in their natural order, their X-coordinates are monotonically increasing or decreasing.