# 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. Prior work on discontinuities had already been discussed in the 1875 memoir of the French mathematician Jean Gaston Darboux. 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 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). 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 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].}  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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