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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.
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 = = +.
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]
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)
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 ...
For Manning — who will forever be considered one of the greatest pocket-passing quarterbacks in league history — the mounting pressure to finally beat Brady reached a boiling point after an 0 ...
In Cohen forcing (named after Paul Cohen) P is the set of functions from a finite subset of ω 2 × ω to {0,1} and p < q if p ⊇ q. This poset satisfies the countable chain condition. Forcing with this poset adds ω 2 distinct reals to the model; this was the poset used by Cohen in his original proof of the independence of the continuum ...