Is Boundedness a Necessity for Double Integral Proofs?

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SUMMARY

The discussion confirms that if a function f is defined on a rectangle R and its double integral exists, then f must be bounded on R when using the ordinary Riemann or Darboux definitions of integration. However, the assertion does not hold for more generalized definitions of integrals, where boundedness is not a requirement. An example provided is the integral of the function f(x, y) = y/√x over the unit square, which illustrates the necessity of boundedness under specific definitions.

PREREQUISITES
  • Understanding of Riemann and Darboux definitions of integration
  • Familiarity with double integrals and their properties
  • Knowledge of bounded and unbounded functions
  • Basic calculus concepts, including limits and continuity
NEXT STEPS
  • Study the properties of Riemann and Darboux integrals in detail
  • Explore generalized definitions of integrals, such as Lebesgue integration
  • Investigate examples of unbounded functions and their integrals
  • Learn about the implications of boundedness in the context of measure theory
USEFUL FOR

Mathematicians, calculus students, and educators interested in the foundational concepts of integration and the implications of boundedness in double integrals.

engin
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Show that if f is defined on a rectangle R and double integral of f on R
exists, then f is necessarily bounded on R.
 
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If the function is defined on the rectangle the I can't see how it could be unbounded there. Do you mean almost everywhere defined? In the second case, the statement is wrong. Try

[tex]\int_0^1\int_0^1 \frac{y}{\sqrt{x}}dxdy[/tex]
 
engin said:
Show that if f is defined on a rectangle R and double integral of f on R
exists, then f is necessarily bounded on R.

This will be true if you define the integral in terms of ordinary Riemann or Darboux definition. There are more general definitions where boundedness is not required.
 

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