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In mathematical analysis, semicontinuity (or semi-continuity) is a property of extended real-valued functions that is weaker than continuity.An extended real-valued function is upper (respectively, lower) semicontinuous at a point if, roughly speaking, the function values for arguments near are not much higher (respectively, lower) than ().
In mathematics, upper hemicontinuity and lower hemicontinuity are extensions of the notions of upper and lower semicontinuity of single-valued functions to set-valued functions. A set-valued function that is both upper and lower hemicontinuous is said to be continuous in an analogy to the property of the same name for single-valued functions.
The function () = + defined for all real numbers is Lipschitz ... Let F(x) be an upper semi-continuous function of x, and that F(x) is a closed, convex set for all x.
In mathematics, the floor function is the function that takes as input a real number x, and gives as output the greatest integer less than or equal to x, denoted ⌊x⌋ or floor(x). Similarly, the ceiling function maps x to the least integer greater than or equal to x , denoted ⌈ x ⌉ or ceil( x ) .
Repeated application of the recurrence relation for the lower incomplete gamma function leads to the power series expansion: [2] (,) = = (+) (+) = = (+ +). Given the rapid growth in absolute value of Γ(z + k) when k → ∞, and the fact that the reciprocal of Γ(z) is an entire function, the coefficients in the rightmost sum are well-defined, and locally the sum converges uniformly for all ...
The definitions can be generalized to functions and even to sets of functions. Given a function f with domain D and a preordered set (K, ≤) as codomain, an element y of K is an upper bound of f if y ≥ f (x) for each x in D. The upper bound is called sharp if equality holds for at least one value of x. It indicates that the constraint is ...
Subharmonic functions make a convex cone, that is, a linear combination of subharmonic functions with positive coefficients is also subharmonic. The pointwise maximum of two subharmonic functions is subharmonic. If the pointwise maximum of a countable number of subharmonic functions is upper semi-continuous, then it is also subharmonic.
The extreme value theorem was originally proven by Bernard Bolzano in the 1830s in a work Function Theory but the work remained unpublished until 1930. Bolzano's proof consisted of showing that a continuous function on a closed interval was bounded, and then showing that the function attained a maximum and a minimum value.