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  2. 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 = = +.

  3. Discontinuities of monotone functions - Wikipedia

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

    As explained in Riesz & Sz.-Nagy (1990), every non-decreasing non-negative function F can be decomposed uniquely as a sum of a jump function f and a continuous monotone function g: the jump function f is constructed by using the jump data of the original monotone function F and it is easy to check that g = F − f is continuous and monotone. [10]

  4. Pointwise convergence - Wikipedia

    en.wikipedia.org/wiki/Pointwise_convergence

    The pointwise limit of a sequence of continuous functions may be a discontinuous function, but only if the convergence is not uniform. For example, f ( x ) = lim n → ∞ cos ⁡ ( π x ) 2 n {\displaystyle f(x)=\lim _{n\to \infty }\cos(\pi x)^{2n}} takes the value 1 {\displaystyle 1} when x {\displaystyle x} is an integer and 0 {\displaystyle ...

  5. Nowhere continuous function - Wikipedia

    en.wikipedia.org/wiki/Nowhere_continuous_function

    In mathematics, a nowhere continuous function, also called an everywhere discontinuous function, is a function that is not continuous at any point of its domain.If is a function from real numbers to real numbers, then is nowhere continuous if for each point there is some > such that for every >, we can find a point such that | | < and | () |.

  6. Thomae's function - Wikipedia

    en.wikipedia.org/wiki/Thomae's_function

    A natural follow-up question one might ask is if there is a function which is continuous on the rational numbers and discontinuous on the irrational numbers. This turns out to be impossible. The set of discontinuities of any function must be an F σ set. If such a function existed, then the irrationals would be an F σ set.

  7. Desmos - Wikipedia

    en.wikipedia.org/wiki/Desmos

    Some 50 employees joined Amplify. Desmos Studio was spun off as a separate public benefit corporation focused on building calculator products and other math tools. [7] In May 2023, Desmos released a beta for a remade Geometry Tool. In it, geometrical shapes can be made, as well as expressions from the normal graphing calculator, with extra ...

  8. 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.

  9. Domain coloring - Wikipedia

    en.wikipedia.org/wiki/Domain_coloring

    Domain coloring plot of the function f(x) = ⁠ (x 2 − 1)(x − 2 − i) 2 / x 2 + 2 + 2i ⁠, using the structured color function described below.. In complex analysis, domain coloring or a color wheel graph is a technique for visualizing complex functions by assigning a color to each point of the complex plane.