Results 261 to 270 of about 127,882 (296)

A theorem on the riemann integral

Indagationes Mathematicae (Proceedings), 1952
\(f(x)\) sei reell, \(-\infty < x < + \infty\). Für alle \(h\) sei \(\int_{-\infty}^{+\infty} |f(x+h)-f(x)| dx=0\). Dann gibt es ein \(c\), so daß \(\int_{-\infty}^{+\infty} |f(x)-c| dx=0.\) Alle auftretenden Integrale sind Riemann-Integrale. Für diesen Satz, den de Bruijn vermutet hat, gibt der Verf. einen interessanten Beweis, für dessen Methode man \
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The Riemann Integral

2018
This chapter develops results concerning the Riemann integration of one variable real functions. The standard necessary and sufficient condition of zero Lebesgue measure for the set of discontinuities for the Riemann integrability of a one variable real function is addressed in detail.
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The Riemann Integral II [PDF]

, 1983
We now consider the legitimacy of passing to the limit under the integral sign. If the sequence 〈f n 〉 of R-integrable functions converges to a limit f on an interval [a, b] does it necessarily follow that $$_{n \to \, + \,\infty }^{\lim }\int_a^b {{f_n}\left( x \right)dx = \int_a^b {f\left( x \right)dx?} }$$ (1.1)
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Repeated Riemann integrals

Mathematical Proceedings of the Cambridge Philosophical Society, 1974
Let f(x, y) be a bounded real function on the unit square. Consider the Riemann integrals (if they exist)
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The Riemann Integral

1998
The definite integral ∫ba(x)dx represents the area under the graph of the function y = f(x) between x = a and x = b. The standard way to define this is to partition the interval [a, b] into a finite number of subintervals, approximate the desired area by sums of areas of rectangles based on these subintervals, and then take the limit as the number of ...
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The Riemann Integral

1973
It is common, at an elementary level, to define integration as the process that is the inverse of differentiation. Thus, given a function f, we look for a function F such that F′(x) = f(x) for all x. There is, in fact, no reason why such a function should exist. Next, if a and b are real numbers with a < b, it is shown that F(b)−F(a) gives the value of
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Riemann’s Integral

2017
This chapter completes the task of establishing the fundamentals of Calculus, this time studying the operation of integration on functions. As we shall see in the next section, in its most simple form this reduces to the computation of areas under the graphs of nonnegative continuous functions \(f: [a,b] \rightarrow \mathbb{R}\), suggesting that ...
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The Riemann Integral

1995
In this chapter we study elementary integration theory for functions defined on closed intervals. Although we expect that the reader has had experience with integral calculus and that the ideas are familiar, we shall not require any special results to be known. For pedagogical reasons we shall first treat the Cauchy integral.
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Riemann Integration

2014
Chapter 9 is about the Riemann integration. It starts with the definition of proper Riemann integral and presents Darboux and Riemann criteria for integrability. It is proved that continuous functions and functions of bounded variation on closed bounded intervals are Riemann integrable.
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An overview of real‐world data sources for oncology and considerations for research

Ca-A Cancer Journal for Clinicians, 2022
Lynne Penberthy, Donna R Rivera, Jennifer L Lund, Msph   +2 more
exaly