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  2. Nowhere continuous function - Wikipedia

    en.wikipedia.org/wiki/Nowhere_continuous_function

    Blumberg theorem – even if a real function : is nowhere continuous, there is a dense subset of such that the restriction of to is continuous. Thomae's function (also known as the popcorn function) – a function that is continuous at all irrational numbers and discontinuous at all rational numbers.

  3. 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 .

  4. 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.

  5. Dirichlet function - Wikipedia

    en.wikipedia.org/wiki/Dirichlet_function

    The Dirichlet function can be constructed as the double pointwise limit of a sequence of continuous functions, as follows: , = ((⁡ (!))) for integer j and k. This shows that the Dirichlet function is a Baire class 2 function.

  6. Absolute continuity - Wikipedia

    en.wikipedia.org/wiki/Absolute_continuity

    The sum and difference of two absolutely continuous functions are also absolutely continuous. If the two functions are defined on a bounded closed interval, then their product is also absolutely continuous. [4] If an absolutely continuous function is defined on a bounded closed interval and is nowhere zero then its reciprocal is absolutely ...

  7. Differentiable function - Wikipedia

    en.wikipedia.org/wiki/Differentiable_function

    The absolute value function is continuous (i.e. it has no gaps). It is differentiable everywhere except at the point x = 0, where it makes a sharp turn as it crosses the y-axis. A cusp on the graph of a continuous function. At zero, the function is continuous but not differentiable. If f is differentiable at a point x 0, then f must also be ...

  8. Pathological (mathematics) - Wikipedia

    en.wikipedia.org/wiki/Pathological_(mathematics)

    A classic example of a pathology is the Weierstrass function, a function that is continuous everywhere but differentiable nowhere. [1] The sum of a differentiable function and the Weierstrass function is again continuous but nowhere differentiable; so there are at least as many such functions as differentiable functions.

  9. 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. Though it is continuous everywhere and has zero ...