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  2. Continuous function - Wikipedia

    en.wikipedia.org/wiki/Continuous_function

    the sinc-function becomes a continuous function on all real numbers. The term removable singularity is used in such cases when (re)defining values of a function to coincide with the appropriate limits make a function continuous at specific points. A more involved construction of continuous functions is the function composition.

  3. Common fixed point problem - Wikipedia

    en.wikipedia.org/wiki/Common_fixed_point_problem

    Boyce extended the work of Maxfield/Mourant and Chu/Moyer in 1971, proving that under some circumstances, commuting functions can have a common fixed point even if one of the functions has period 2 fixed points. [23] His work was later extended by Theodore Mitchell, Julio Cano, and Jacek R. Jachymski. [24] [25] [26]

  4. Classification of discontinuities - Wikipedia

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

    Continuous functions are of utmost importance in mathematics, functions and applications. However, not all functions are continuous. If a function is not continuous at a limit point (also called "accumulation point" or "cluster point") of its domain , one says that it has a discontinuity there.

  5. Complete metric space - Wikipedia

    en.wikipedia.org/wiki/Complete_metric_space

    The space C [a, b] of continuous real-valued functions on a closed and bounded interval is a Banach space, and so a complete metric space, with respect to the supremum norm. However, the supremum norm does not give a norm on the space C (a, b) of continuous functions on (a, b), for it may contain unbounded functions.

  6. Fixed-point theorem - Wikipedia

    en.wikipedia.org/wiki/Fixed-point_theorem

    The Banach fixed-point theorem (1922) gives a general criterion guaranteeing that, if it is satisfied, the procedure of iterating a function yields a fixed point. [2]By contrast, the Brouwer fixed-point theorem (1911) is a non-constructive result: it says that any continuous function from the closed unit ball in n-dimensional Euclidean space to itself must have a fixed point, [3] but it doesn ...

  7. Uniform continuity - Wikipedia

    en.wikipedia.org/wiki/Uniform_continuity

    The Heine–Cantor theorem asserts that every continuous function on a compact set is uniformly continuous. In particular, if a function is continuous on a closed bounded interval of the real line, it is uniformly continuous on that interval. The Darboux integrability of continuous functions follows almost immediately from this theorem.

  8. Closed graph property - Wikipedia

    en.wikipedia.org/wiki/Closed_graph_property

    Definition: We say that the function (resp. set-valued function) f is closable in X × Y if there exists a subset D ⊆ X containing S and a function (resp. set-valued function) F : D → Y whose graph is equal to the closure of the set Gr f in X × Y. Such an F is called a closure of f in X × Y, is denoted by f, and necessarily extends f.

  9. Liouville's theorem (complex analysis) - Wikipedia

    en.wikipedia.org/wiki/Liouville's_theorem...

    By the extreme value theorem, a continuous function on a closed and bounded set obtains its extreme values, implying that / | | for some constant and ¯ (,). Thus, the function q ( z ) {\displaystyle q(z)} is bounded in C {\displaystyle \mathbb {C} } , and by Liouville's theorem, is constant , which contradicts our assumption that p ...