# Discontinuités des fonctions monotones

Généralement, 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.[1] Prior work on discontinuities had already been discussed in the 1875 memoir of the French mathematician Jean Gaston Darboux.[2] Contenu 1 Définitions 2 Precise statement 3 Preuves 3.1 Proof when the domain is closed and bounded 3.1.1 Preuve 1 3.1.2 Preuve 2 3.2 Proof of general case 4 Jump functions 5 Voir également 6 Remarques 7 Références 8 Bibliography Definitions Denote the limit from the left by {style d'affichage(x^{-}droit):=lim _{znearrow x}F(z)=lim _{empiler {hto 0}{h>0}}F(x-h)} and denote the limit from the right by {style d'affichage(x^{+}droit):=lim _{zsearrow x}F(z)=lim _{empiler {hto 0}{h>0}}F(x+h).} Si {style d'affichage(x^{+}droit)} et {style d'affichage(x^{-}droit)} exist and are finite then the difference {style d'affichage(x^{+}droit)-volé(x^{-}droit)} is called the jump[3] de {style d'affichage f} à {displaystyle x.} Consider a real-valued function {style d'affichage f} of real variable {style d'affichage x} defined in a neighborhood of a point {displaystyle x.} Si {style d'affichage f} is discontinuous at the point {style d'affichage x} then the discontinuity will be a removable discontinuity, or an essential discontinuity, or a jump discontinuity (also called a discontinuity of the first kind).[4] If the function is continuous at {style d'affichage x} then the jump at {style d'affichage x} est zéro. En outre, si {style d'affichage f} is not continuous at {style d'affichage x,} the jump can be zero at {style d'affichage x} si {style d'affichage(x^{+}droit)= vol(x^{-}droit)neq f(X).} Precise statement Let {style d'affichage f} be a real-valued monotone function defined on an interval {displaystyle I.} 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, du premier genre. With this remark the theorem takes the stronger form: Laisser {style d'affichage f} be a monotone function defined on an interval {displaystyle I.} Then the set of discontinuities is at most countable.

Proofs This proof starts by proving the special case where the function's domain is a closed and bounded interval {style d'affichage [un,b].} [6][7] The proof of the general case follows from this special case.

Proof when the domain is closed and bounded Two proofs of this special case are given.

Preuve 1 Laisser {style d'affichage I:=[un,b]} be an interval and let {style d'affichage f:Ito mathbb {R} } be a non-decreasing function (such as an increasing function). Then for any {style d'affichage a0} et laissez {style d'affichage x_{1}while if {displaystyle fsearrow } then pick {style d'affichage y_{ré}en mathbb {Q} } pour que {style d'affichage(d^{-}droit)>y_{ré}>fleft(d^{+}droit)} détient).

It will now be shown that if {displaystyle d,ein D} sont distincts, say with {displaystyle dvolé(d^{+}droit)geq fleft(e ^{-}droit)>y_{e}.} Dans les deux cas, {style d'affichage y_{ré}neq y_{e}.} Thus every {displaystyle din D} is associated with a unique rational number (said differently, the map {displaystyle Dto mathbb {Q} } Défini par {displaystyle dmapsto y_{ré}} est injectif). Depuis {style d'affichage mathbb {Q} } is countable, the same must be true of {displaystyle D.} {displaystyle blacksquare } Proof of general case Suppose that the domain of {style d'affichage f} (a monotone real-valued function) is equal to a union of countably many closed and bounded intervals; say its domain is {style d'affichage bigcup _{n}la gauche[un_{n},b_{n}droit]} (no requirements are placed on these closed and bounded intervals[un]). It follows from the special case proved above that for every index {displaystyle n,} the restriction {style d'affichage f{big vert }_{la gauche[un_{n},b_{n}droit]}:la gauche[un_{n},b_{n}droit]à mathbb {R} } de {style d'affichage f} to the interval {style d'affichage à gauche[un_{n},b_{n}droit]} has at most countably many discontinuities; denote this (countable) set of discontinuities by {displaystyle D_{n}.} Si {style d'affichage f} has a discontinuity at a point {style d'affichage x_{0}in bigcup _{n}la gauche[un_{n},b_{n}droit]} in its domain then either {style d'affichage x_{0}} is equal to an endpoint of one of these intervals (C'est, {style d'affichage x_{0}in left{un_{1},b_{1},un_{2},b_{2},ldots right}} ) or else there exists some index {displaystyle n} tel que {style d'affichage a_{n}must be a point of discontinuity for {style d'affichage f{big vert }_{la gauche[un_{n},b_{n}droit]}} (C'est, {style d'affichage x_{0}in D_{n}} ). Thus the set {displaystyle D} of all points of at which {style d'affichage f} is discontinuous is a subset of {style d'affichage à gauche{un_{1},b_{1},un_{2},b_{2},ldots right}cup bigcup _{n}RÉ_{n},} which is a countable set (because it is a union of countably many countable sets) so that its subset {displaystyle D} must also be countable (because every subset of a countable set is countable). En particulier, because every interval (including open intervals and half open/closed intervals) of real numbers can be written as a countable union of closed and bounded intervals, it follows that any monotone real-valued function defined on an interval has at most countable many discontinuities. To make this argument more concrete, suppose that the domain of {style d'affichage f} is an interval {style d'affichage I} that is not closed and bounded (and hence by Heine–Borel theorem not compact). Then the interval can be written as a countable union of closed and bounded intervals {style d'affichage I_{n}} with the property that any two consecutive intervals have an endpoint in common: {displaystyle I=cup _{n=1}^{infime }JE_{n}.} Si {style d'affichage I=(un,b]{texte{ avec }}ageq -infty } alors {style d'affichage I_{1}=gauche[Alpha _{1},bright], JE_{2}=gauche[Alpha _{2},Alpha _{1}droit],ldots ,JE_{n}=gauche[Alpha _{n},Alpha _{n-1}droit],ldots } où {style d'affichage à gauche(Alpha _{n}droit)_{n=1}^{infime }} is a strictly decreasing sequence such that {style d'affichage alpha _{n}rightarrow a.} In a similar way if {style d'affichage I=[un,b),{texte{ avec }}bleq +infty } or if {style d'affichage I=(un,b){texte{ avec }}-infty leq a 0 for each n. Définir {style d'affichage f_{n}(X)=0,,} pour {style d'affichage ,,XX_{n}.} Then the jump function, or saltus-function, Défini par {style d'affichage f(X)=,,somme _{n=1}^{infime }F_{n}(X)=,,somme _{X_{n}leq x}lambda _{n}+somme _{X_{n}

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