About: Discontinuities of monotone functions

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In the mathematical field of analysis, a well-known theorem describes the set of discontinuities of a monotone real-valued function of a real variable; all discontinuities of such a (monotone) function are necessarily jump discontinuities and there are at most countably many of them.

Attributes	Values
rdfs:label	Discontinuities of monotone functions (en) Théorème de Froda (fr)
rdfs:comment	En analyse réelle, le théorème de Froda, redécouvert en 1929 par le mathématicien roumain Alexandru Froda mais dont des versions plus générales avaient été trouvées de 1907 à 1910 par Grace Chisholm Young et William Henry Young, assure que l'ensemble des points de discontinuité de première espèce d'une fonction réelle d'une variable réelle (définie sur un intervalle) est au plus dénombrable. (fr) In the mathematical field of analysis, a well-known theorem describes the set of discontinuities of a monotone real-valued function of a real variable; all discontinuities of such a (monotone) function are necessarily jump discontinuities and there are at most countably many of them. (en)
dcterms:subject	Articles containing proofs Theorems in real analysis Continuous mappings
Wikipage page ID	22278053 (xsd:integer)
Wikipage revision ID	1097505803 (xsd:integer)
Link from a Wikipage to another Wikipage	Cambridge University Press Princeton University Press Proc. London Math. Soc. Monotonic function Real-valued function Almost everywhere John Wiley & Sons Articles containing proofs Countable set Mathematical analysis Mathematics Bounded set Closed set Empty set Step function Disjoint sets Heine–Borel theorem Addison-Wesley Alexandru Froda Null set Discontinuity (mathematics) Removable discontinuity Henri Lebesgue Interval (mathematics) Jean Gaston Darboux Theorems in real analysis Continuous mappings Dini derivative Indicator function Injective function Rational number Sequence Mathematical Association of America Jump discontinuity Limit from the left Limit from the right Springer-Verlag Restriction of a map Essential discontinuity Compact set Monotone function
Link from a Wikipage to an external page	https://londmathsoc.onlinelibrary.wiley.com/doi/epdf/10.1112/plms/s2-9.1.325%7Cfirst1=William https://archive.org/details/in.ernet.dli.2015.141035/page/n173/mode/2up https://archive.org/details/introductiontoth0000ojas https://archive.org/details/lebesgueintegral0000burk https://archive.org/details/theoryfunctions00hobsgoog/page/244/mode/2up%7C https://archive.org/details/theoryoffunction00nat%7Cmr=0067952 https://books.google.com/books%3Fid=D_XBAgAAQBAJ&pg=PA28 https://www.cambridge.org/core/books/primer-of-real-functions/097C383F0BF28D65F3D32D9A9924548B http://matwbn.icm.edu.pl/ksiazki/cm/cm10/cm10138.pdf http://matwbn.icm.edu.pl/ksiazki/cm/cm8/cm8115.pdf%7Cmr=0126513 http://matwbn.icm.edu.pl/ksiazki/cm/cm8/cm8136.pdf%7Cmr=0158036%7Clast=Lipi%C5%84ski%7Cfirst= http://matwbn.icm.edu.pl/ksiazki/mon/mon07/mon0703.pdf
sameAs	Discontinuities of monotone functions Discontinuities of monotone functions Discontinuities of monotone functions
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left	yes (en)
title	Proof that a jump function has zero derivative almost everywhere. (en)
has abstract	In the mathematical field of analysis, a well-known theorem describes the set of discontinuities of a monotone real-valued function of a real variable; all discontinuities of such a (monotone) function are necessarily jump discontinuities and there are at most countably many of them. Usually, this theorem appears in literature without a name. It is called Froda's theorem in some recent works; in his 1929 dissertation, Alexandru Froda stated that the result was previously well-known and had provided his own elementary proof for the sake of convenience. Prior work on discontinuities had already been discussed in the 1875 memoir of the French mathematician Jean Gaston Darboux. (en) En analyse réelle, le théorème de Froda, redécouvert en 1929 par le mathématicien roumain Alexandru Froda mais dont des versions plus générales avaient été trouvées de 1907 à 1910 par Grace Chisholm Young et William Henry Young, assure que l'ensemble des points de discontinuité de première espèce d'une fonction réelle d'une variable réelle (définie sur un intervalle) est au plus dénombrable. (fr)
prov:wasDerivedFrom	wikipedia-en:Discontinuities_of_monotone_functions?oldid=1097505803&ns=0
page length (characters) of wiki page	28025 (xsd:nonNegativeInteger)
foaf:isPrimaryTopicOf	wikipedia-en:Discontinuities_of_monotone_functions
is Link from a Wikipage to another Wikipage of	Monotonic function Measure (mathematics) Froda's theorem
is Wikipage redirect of	Froda's theorem
is foaf:primaryTopic of	wikipedia-en:Discontinuities_of_monotone_functions

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