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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. Uniform continuity - Wikipedia

    en.wikipedia.org/wiki/Uniform_continuity

    Linear functions + are the simplest examples of uniformly continuous functions. Any continuous function on the interval [ 0 , 1 ] {\displaystyle [0,1]} is also uniformly continuous, since [ 0 , 1 ] {\displaystyle [0,1]} is a compact set.

  4. Closed graph theorem - Wikipedia

    en.wikipedia.org/wiki/Closed_graph_theorem

    An example of non-compact is the real line, which allows the discontinuous function with closed graph () = {,. Also, closed linear operators in functional analysis (linear operators with closed graphs) are typically not continuous.

  5. Lipschitz continuity - Wikipedia

    en.wikipedia.org/wiki/Lipschitz_continuity

    A function is called locally Lipschitz continuous if for every x in X there exists a neighborhood U of x such that f restricted to U is Lipschitz continuous. Equivalently, if X is a locally compact metric space, then f is locally Lipschitz if and only if it is Lipschitz continuous on every compact subset of X .

  6. Weierstrass function - Wikipedia

    en.wikipedia.org/wiki/Weierstrass_function

    In mathematics, the Weierstrass function, named after its discoverer, Karl Weierstrass, is an example of a real-valued function that is continuous everywhere but differentiable nowhere. It is also an example of a fractal curve .

  7. Intermediate value theorem - Wikipedia

    en.wikipedia.org/wiki/Intermediate_value_theorem

    The intermediate value theorem says that every continuous function is a Darboux function. However, not every Darboux function is continuous; i.e., the converse of the intermediate value theorem is false. As an example, take the function f : [0, ∞) → [−1, 1] defined by f(x) = sin(1/x) for x > 0 and f(0) = 0.

  8. 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 ...

  9. Absolute continuity - Wikipedia

    en.wikipedia.org/wiki/Absolute_continuity

    A continuous function fails to be absolutely continuous if it fails to be uniformly continuous, which can happen if the domain of the function is not compact – examples are tan(x) over [0, π/2), x 2 over the entire real line, and sin(1/x) over (0, 1]. But a continuous function f can