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  2. Discontinuities of monotone functions - Wikipedia

    en.wikipedia.org/wiki/Discontinuities_of...

    Let be a real-valued monotone function defined on an interval. Then the set of discontinuities of the first kind is at most countable.. One can prove [5] [3] that all points of discontinuity of a monotone real-valued function defined on an interval are jump discontinuities and hence, by our definition, of the first kind.

  3. Helly's selection theorem - Wikipedia

    en.wikipedia.org/wiki/Helly's_selection_theorem

    Let X be a reflexive, separable Hilbert space and let E be a closed, convex subset of X. Let Δ : X → [0, +∞) be positive-definite and homogeneous of degree one. Suppose that z n is a uniformly bounded sequence in BV([0, T]; X) with z n (t) ∈ E for all n ∈ N and t ∈ [0, T].

  4. Classification of discontinuities - Wikipedia

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

    The function in example 1, a removable discontinuity. Consider the piecewise function = {< = >. The point = is a removable discontinuity.For this kind of discontinuity: The one-sided limit from the negative direction: = and the one-sided limit from the positive direction: + = + at both exist, are finite, and are equal to = = +.

  5. Thomae's function - Wikipedia

    en.wikipedia.org/wiki/Thomae's_function

    It is named after Carl Johannes Thomae, but has many other names: the popcorn function, the raindrop function, the countable cloud function, the modified Dirichlet function, the ruler function (not to be confused with the integer ruler function), [2] the Riemann function, or the Stars over Babylon (John Horton Conway's name). [3]

  6. Talk:Discontinuities of monotone functions - Wikipedia

    en.wikipedia.org/wiki/Talk:Discontinuities_of...

    You say "Let x 1 < x 2 < x 3 < ⋅⋅⋅ be a countable subset of the compact interval [a, b] ..." as if every countable set of reals can be put in this form. That is not so. Although the rational numbers is countable, it cannot be enumerated in a strictly increasing sequence. JRSpriggs 02:32, 9 February 2022 (UTC)

  7. Cantor function - Wikipedia

    en.wikipedia.org/wiki/Cantor_function

    The graph of the Cantor function on the unit interval. In mathematics, the Cantor function is an example of a function that is continuous, but not absolutely continuous.It is a notorious counterexample in analysis, because it challenges naive intuitions about continuity, derivative, and measure.

  8. Which foods are considered 'healthy?' FDA issues new label ...

    www.aol.com/news/foods-considered-healthy-fda...

    The Food and Drug Administration's new rules on "healthy" food labels are voluntary and are scheduled to take effect at the end of February.

  9. Talk:Monotonic function - Wikipedia

    en.wikipedia.org/wiki/Talk:Monotonic_function

    From Salas and Hille, Calculus of One and Several Variables, 1982, Section 4.2 "Increasing and Decreasing Functions", Definition 4.2.1, p. 142: "A function f is said to (i) increase on the interval I iff for every two numbers x 1, x 2 in I x 1 < x 2 implies f(x 1) < f(x 2).