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  2. Continuous function - Wikipedia

    en.wikipedia.org/wiki/Continuous_function

    Continuity is one of the core concepts of calculus and mathematical analysis, where arguments and values of functions are real and complex numbers. The concept has been generalized to functions between metric spaces and between topological spaces .

  3. Absolute continuity - Wikipedia

    en.wikipedia.org/wiki/Absolute_continuity

    In calculus and real analysis, absolute continuity is a smoothness property of functions that is stronger than continuity and uniform continuity. The notion of absolute continuity allows one to obtain generalizations of the relationship between the two central operations of calculus — differentiation and integration .

  4. List of continuity-related mathematical topics - Wikipedia

    en.wikipedia.org/wiki/List_of_continuity-related...

    Continuous function; Absolutely continuous function; Absolute continuity of a measure with respect to another measure; Continuous probability distribution: Sometimes this term is used to mean a probability distribution whose cumulative distribution function (c.d.f.) is (simply) continuous.

  5. Smoothness - Wikipedia

    en.wikipedia.org/wiki/Smoothness

    For example, reflections in a car body will not appear smooth unless the body has continuity. [ citation needed ] A rounded rectangle (with ninety degree circular arcs at the four corners) has G 1 {\displaystyle G^{1}} continuity, but does not have G 2 {\displaystyle G^{2}} continuity.

  6. Continuous functional calculus - Wikipedia

    en.wikipedia.org/wiki/Continuous_functional_calculus

    If one wants to extend the natural functional calculus for polynomials on the spectrum of an element of a Banach algebra to a functional calculus for continuous functions (()) on the spectrum, it seems obvious to approximate a continuous function by polynomials according to the Stone-Weierstrass theorem, to insert the element into these polynomials and to show that this sequence of elements ...

  7. Limit of a function - Wikipedia

    en.wikipedia.org/wiki/Limit_of_a_function

    Although implicit in the development of calculus of the 17th and 18th centuries, the modern idea of the limit of a function goes back to Bolzano who, in 1817, introduced the basics of the epsilon-delta technique (see (ε, δ)-definition of limit below) to define continuous functions.

  8. Intermediate value theorem - Wikipedia

    en.wikipedia.org/wiki/Intermediate_value_theorem

    Intermediate value theorem: Let be a continuous function defined on [,] and let be a number with () < < ().Then there exists some between and such that () =.. In mathematical analysis, the intermediate value theorem states that if is a continuous function whose domain contains the interval [a, b], then it takes on any given value between () and () at some point within the interval.

  9. Modulus of continuity - Wikipedia

    en.wikipedia.org/wiki/Modulus_of_continuity

    A sublinear modulus of continuity can easily be found for any uniformly continuous function which is a bounded perturbation of a Lipschitz function: if f is a uniformly continuous function with modulus of continuity ω, and g is a k Lipschitz function with uniform distance r from f, then f admits the sublinear module of continuity min{ω(t), 2r ...

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