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Let be a real-valued monotone function defined on an interval. Then the set of discontinuities of the first kind is at most countable.. One can prove [5] [3] that all points of discontinuity of a monotone real-valued function defined on an interval are jump discontinuities and hence, by our definition, of the first kind.
However, furtherly, both authors state that is always a countable set (see [6] [7]). The term essential discontinuity has evidence of use in mathematical context as early as 1889. [8] However, the earliest use of the term alongside a mathematical definition seems to have been given in the work by John Klippert. [9]
An increasing function f on an interval I has at most countably many points of discontinuity. 2.2 Step 2. Inductive Construction of a subsequence converging at discontinuities and rationals.
It is named after Carl Johannes Thomae, but has many other names: the popcorn function, the raindrop function, the countable cloud function, the modified Dirichlet function, the ruler function (not to be confused with the integer ruler function), [2] the Riemann function, or the Stars over Babylon (John Horton Conway's name). [3]
But if you turn it on its side, don't you get a function (okay, specifying the values on the straight line bits in some arbitrary way) from [0, 1] to [0, 1] which is monotonic and which has an uncountable number of discontinuities? -- Oliver Pereira 02:27 Nov 29, 2002 (UTC) Nope, okay, the set of discontinuities is countable, of course. Why don ...
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You say "Let x 1 < x 2 < x 3 < ⋅⋅⋅ be a countable subset of the compact interval [a, b] ..." as if every countable set of reals can be put in this form. That is not so. Although the rational numbers is countable, it cannot be enumerated in a strictly increasing sequence. JRSpriggs 02:32, 9 February 2022 (UTC)
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