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  2. Riemann–Lebesgue lemma - Wikipedia

    en.wikipedia.org/wiki/Riemann–Lebesgue_lemma

    A version holds for Fourier series as well: if is an integrable function on a bounded interval, then the Fourier coefficients ^ of tend to 0 as . This follows by extending f {\displaystyle f} by zero outside the interval, and then applying the version of the Riemann–Lebesgue lemma on the entire real line.

  3. Riemann integral - Wikipedia

    en.wikipedia.org/wiki/Riemann_integral

    A bounded function on a compact interval [a, b] is Riemann integrable if and only if it is continuous almost everywhere (the set of its points of discontinuity has measure zero, in the sense of Lebesgue measure).

  4. Classification of discontinuities - Wikipedia

    en.wikipedia.org/wiki/Classification_of...

    A bounded function, , is Riemann integrable on [,] if and only if the correspondent set of all essential discontinuities of first kind of has Lebesgue's measure zero. The case where E 1 = ∅ {\displaystyle E_{1}=\varnothing } correspond to the following well-known classical complementary situations of Riemann integrability of a bounded ...

  5. Limits of integration - Wikipedia

    en.wikipedia.org/wiki/Limits_of_integration

    of a Riemann integrable function defined on a closed and bounded interval are the real numbers and , in which is called the lower limit and the upper limit. The region that is bounded can be seen as the area inside a {\displaystyle a} and b {\displaystyle b} .

  6. Integral - Wikipedia

    en.wikipedia.org/wiki/Integral

    Although all bounded piecewise continuous functions are Riemann-integrable on a bounded interval, subsequently more general functions were considered—particularly in the context of Fourier analysis—to which Riemann's definition does not apply, and Lebesgue formulated a different definition of integral, founded in measure theory (a subfield ...

  7. Thomae's function - Wikipedia

    en.wikipedia.org/wiki/Thomae's_function

    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.

  8. Riemann–Hilbert problem - Wikipedia

    en.wikipedia.org/wiki/Riemann–Hilbert_problem

    Given an oriented contour (technically: an oriented union of smooth curves without points of infinite self-intersection in the complex plane), a Riemann–Hilbert factorization problem is the following. Given a matrix function () defined on the contour , find a holomorphic matrix function () defined on the complement of , such that the ...

  9. Real analysis - Wikipedia

    en.wikipedia.org/wiki/Real_analysis

    A function is Darboux integrable if the upper and lower Darboux sums can be made to be arbitrarily close to each other for a sufficiently small mesh. Although this definition gives the Darboux integral the appearance of being a special case of the Riemann integral, they are, in fact, equivalent, in the sense that a function is Darboux ...