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The standard part function can also be defined for infinite hyperreal numbers as follows: If x is a positive infinite hyperreal number, set st(x) to be the extended real number +, and likewise, if x is a negative infinite hyperreal number, set st(x) to be (the idea is that an infinite hyperreal number should be smaller than the "true" absolute ...
A similar but different real-line system, the projectively extended real line, does not distinguish between + and (i.e. infinity is unsigned). [4] As a result, a function may have limit on the projectively extended real line, while in the extended real number system only the absolute value of the function has a limit, e.g. in the case of the ...
The image of a function f(x 1, x 2, …, x n) is the set of all values of f when the n-tuple (x 1, x 2, …, x n) runs in the whole domain of f.For a continuous (see below for a definition) real-valued function which has a connected domain, the image is either an interval or a single value.
In mathematics, a real-valued function is a function whose values are real numbers. In other words, it is a function that assigns a real number to each member of its domain . Real-valued functions of a real variable (commonly called real functions ) and real-valued functions of several real variables are the main object of study of calculus and ...
The converse, though, does not necessarily hold: for example, taking f as =, where V is a Vitali set, it is clear that f is not measurable, but its absolute value is, being a constant function. The positive part and negative part of a function are used to define the Lebesgue integral for a real-valued function.
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 convex analysis and variational analysis, a point (in the domain) at which some given function is minimized is typically sought, where is valued in the extended real number line [,] = {}. [1] Such a point, if it exists, is called a global minimum point of the function and its value at this point is called the global minimum (value) of the ...
The Lebesgue criterion for integrability states that a bounded function is Riemann integrable if and only if the set of all discontinuities has measure zero. [5] Every countable subset of the real numbers - such as the rational numbers - has measure zero, so the above discussion shows that Thomae's function is Riemann integrable on any interval.