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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.
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.
A function that is absolutely monotonic on [,) can be extended to a function that is not only analytic on the real line but is even the restriction of an entire function to the real line. The big Bernshtein theorem : A function f ( x ) {\displaystyle f(x)} that is absolutely monotonic on ( − ∞ , 0 ] {\displaystyle (-\infty ,0]} can be ...
The term removable discontinuity is sometimes broadened to include a removable singularity, in which the limits in both directions exist and are equal, while the function is undefined at the point . [a] This use is an abuse of terminology because continuity and discontinuity of a function are concepts defined only for points in the function's ...
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 Eduard ...
In this way any monotone function can be written in a unique way as the sum of a continuous monotone function and a jump function. Since the formula for H ( x ) {\displaystyle H(x)} is a positive combination of characteristic functions, it is a uniformly convergent sum, so the analysis of Riesz & Sz.-Nagy (1990 , pp. 13–15) is particularly ...
By the way, I'd say the first sentence in the Section "Monotonicity in calculus and analysis" is confusing. That is the definition of an increasing (or non-decreasing) function. A monotone function may be either increasing or decreasing (as defined in the next sentence). At least that is standard terminology in calculus textbooks.
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 ).