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Say (,) is equipped with its usual topology. Then the essential range of f is given by . = { >: < {: | | <}}. [7]: Definition 4.36 [8] [9]: cf. Exercise 6.11 In other words: The essential range of a complex-valued function is the set of all complex numbers z such that the inverse image of each ε-neighbourhood of z under f has positive measure.
As is often the case in measure-theoretic questions, the definition of essential supremum and infimum does not start by asking what a function does at points (that is, the image of ), but rather by asking for the set of points where equals a specific value (that is, the preimage of under ).
supremum = least upper bound. A lower bound of a subset of a partially ordered set (,) is an element of such that . for all .; A lower bound of is called an infimum (or greatest lower bound, or meet) of if
The preamble of the CGC 1 states that its function is to provide "direction to the non-Federal Governments (NFGs) for developing continuity plans and programs. Continuity planning facilitates the performance of essential functions during all-hazards emergencies or other situations that may disrupt normal operations.
As a consequence of Liouville's theorem, any function that is entire on the whole Riemann sphere [d] is constant. Thus any non-constant entire function must have a singularity at the complex point at infinity, either a pole for a polynomial or an essential singularity for a transcendental entire function.
For some functions, the image and the codomain coincide; these functions are called surjective or onto. For example, consider the function () =, which inputs a real number and outputs its double. For this function, both the codomain and the image are the set of all real numbers, so the word range is unambiguous.
in an essential discontinuity (a.k.a. infinite discontinuity), oscillation measures the failure of a limit to exist. A special case is if the function diverges to infinity or minus infinity, in which case the oscillation is not defined (in the extended real numbers, this is a removable discontinuity).
In that case, the essential support of a measurable function : written (), is defined to be the smallest closed subset of such that =-almost everywhere outside . Equivalently, is the complement of the largest open set on which =-almost everywhere [5] ():= {: =}.