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  2. Cardinality of the continuum - Wikipedia

    en.wikipedia.org/wiki/Cardinality_of_the_continuum

    Cantor defined cardinality in terms of bijective functions: two sets have the same cardinality if, and only if, there exists a bijective function between them. Between any two real numbers a < b, no matter how close they are to each other, there are always infinitely many other real numbers, and Cantor showed that they are as many as those ...

  3. Function of a real variable - Wikipedia

    en.wikipedia.org/wiki/Function_of_a_real_variable

    This follows from the fact that a continuous function is completely determined by its value on a dense subset of its domain. [2] Thus, the cardinality of the set of continuous real-valued functions on the reals is no greater than the cardinality of the set of real-valued functions of a rational variable. By cardinal arithmetic:

  4. List of continuity-related mathematical topics - Wikipedia

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

    Continuous probability distribution: Sometimes this term is used to mean a probability distribution whose cumulative distribution function (c.d.f.) is (simply) continuous. Sometimes it has a less inclusive meaning: a distribution whose c.d.f. is absolutely continuous with respect to Lebesgue measure. This less inclusive sense is equivalent to ...

  5. Cardinal characteristic of the continuum - Wikipedia

    en.wikipedia.org/wiki/Cardinal_characteristic_of...

    As is standard in set theory, we denote by the least infinite ordinal, which has cardinality ; it may be identified with the set of natural numbers.. A number of cardinal characteristics naturally arise as cardinal invariants for ideals which are closely connected with the structure of the reals, such as the ideal of Lebesgue null sets and the ideal of meagre sets.

  6. Continuum (set theory) - Wikipedia

    en.wikipedia.org/wiki/Continuum_(set_theory)

    The cardinality of the continuum is the size of the set of real numbers. The continuum hypothesis is sometimes stated by saying that no cardinality lies between that of the continuum and that of the natural numbers , ℵ 0 {\displaystyle \aleph _{0}} , or alternatively, that c = ℵ 1 {\displaystyle {\mathfrak {c}}=\aleph _{1}} .

  7. 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 .

  8. Cardinality - Wikipedia

    en.wikipedia.org/wiki/Cardinality

    Bijective function from N to the set E of even numbers. Although E is a proper subset of N, both sets have the same cardinality. N does not have the same cardinality as its power set P(N): For every function f from N to P(N), the set T = {n∈N: n∉f(n)} disagrees with every set in the range of f, hence f cannot be surjective.

  9. Continuum hypothesis - Wikipedia

    en.wikipedia.org/wiki/Continuum_hypothesis

    The continuum hypothesis states that the set of real numbers has minimal possible cardinality which is greater than the cardinality of the set of integers. That is, every set, S, of real numbers can either be mapped one-to-one into the integers or the real numbers can be mapped one-to-one into S.