Discontinuities of monotone functions

Discontinuities of monotone functions (Redirected from Froda's theorem) Jump to navigation Jump to search 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 (monoton) function are necessarily jump discontinuities and there are at most countably many of them.

Normalerweise, 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] Inhalt 1 Definitionen 2 Precise statement 3 Beweise 3.1 Proof when the domain is closed and bounded 3.1.1 Nachweisen 1 3.1.2 Nachweisen 2 3.2 Proof of general case 4 Jump functions 5 Siehe auch 6 Anmerkungen 7 Verweise 8 Bibliography Definitions Denote the limit from the left by {displaystyle fleft(x^{-}Rechts):=lim _{znearrow x}f(z)=lim _{Stapel {hto 0}{h>0}}f(x-h)} and denote the limit from the right by {displaystyle fleft(x^{+}Rechts):=lim _{zsearrow x}f(z)=lim _{Stapel {hto 0}{h>0}}f(x+h).} Wenn {displaystyle fleft(x^{+}Rechts)} und {displaystyle fleft(x^{-}Rechts)} exist and are finite then the difference {displaystyle fleft(x^{+}Rechts)-geflogen(x^{-}Rechts)} is called the jump[3] von {Anzeigestil f} bei {displaystyle x.} Consider a real-valued function {Anzeigestil f} of real variable {Anzeigestil x} defined in a neighborhood of a point {displaystyle x.} Wenn {Anzeigestil f} is discontinuous at the point {Anzeigestil 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 {Anzeigestil x} then the jump at {Anzeigestil x} ist Null. Darüber hinaus, wenn {Anzeigestil f} is not continuous at {Anzeigestil x,} the jump can be zero at {Anzeigestil x} wenn {displaystyle fleft(x^{+}Rechts)= links(x^{-}Rechts)neq f(x).} Precise statement Let {Anzeigestil 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, der ersten Sorte. With this remark the theorem takes the stronger form: Lassen {Anzeigestil 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 {Anzeigestil [a,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.

Nachweisen 1 Lassen {Anzeigestil I:=[a,b]} be an interval and let {Anzeigestil f:Ito mathbb {R} } be a non-decreasing function (such as an increasing function). Then for any {Anzeigestil a0} und lass {Anzeigestil x_{1}while if {displaystyle fsearrow } then pick {Anzeigestil y_{d}in mathbb {Q} } so dass {displaystyle fleft(d^{-}Rechts)>y_{d}>fleft(d^{+}Rechts)} hält).

It will now be shown that if {Anzeigestil d,ein D} sind verschieden, say with {Anzeigestil dgeflogen(d^{+}Rechts)geq fleft(e^{-}Rechts)>y_{e}.} So oder so, {Anzeigestil y_{d}neq y_{e}.} Thus every {displaystyle din D} is associated with a unique rational number (said differently, the map {displaystyle Dto mathbb {Q} } definiert von {displaystyle dmapsto y_{d}} ist injektiv). Seit {Anzeigestil mathbb {Q} } is countable, the same must be true of {displaystyle D.} {Anzeigestil blacksquare } Proof of general case Suppose that the domain of {Anzeigestil f} (a monotone real-valued function) is equal to a union of countably many closed and bounded intervals; say its domain is {displaystyle bigcup _{n}links[a_{n},b_{n}Rechts]} (no requirements are placed on these closed and bounded intervals[a]). It follows from the special case proved above that for every index {Anzeigestil n,} the restriction {Anzeigestil f{big vert }_{links[a_{n},b_{n}Rechts]}:links[a_{n},b_{n}Rechts]zu mathbb {R} } von {Anzeigestil f} to the interval {Anzeigestil links[a_{n},b_{n}Rechts]} has at most countably many discontinuities; denote this (countable) set of discontinuities by {displaystyle D_{n}.} Wenn {Anzeigestil f} has a discontinuity at a point {Anzeigestil x_{0}in bigcup _{n}links[a_{n},b_{n}Rechts]} in its domain then either {Anzeigestil x_{0}} is equal to an endpoint of one of these intervals (das ist, {Anzeigestil x_{0}in left{a_{1},b_{1},a_{2},b_{2},ldots right}} ) or else there exists some index {Anzeigestil n} so dass {Anzeigestil a_{n}must be a point of discontinuity for {Anzeigestil f{big vert }_{links[a_{n},b_{n}Rechts]}} (das ist, {Anzeigestil x_{0}in D_{n}} ). Thus the set {Anzeigestil D} of all points of at which {Anzeigestil f} is discontinuous is a subset of {Anzeigestil links{a_{1},b_{1},a_{2},b_{2},ldots right}cup bigcup _{n}D_{n},} which is a countable set (because it is a union of countably many countable sets) so that its subset {Anzeigestil D} must also be countable (because every subset of a countable set is countable). Im Speziellen, 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 {Anzeigestil f} is an interval {Anzeigestil 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 {Anzeigestil I_{n}} with the property that any two consecutive intervals have an endpoint in common: {displaystyle I=cup _{n=1}^{unendlich }ICH_{n}.} Wenn {Anzeigestil I=(a,b]{Text{ mit }}ageq -infty } dann {Anzeigestil I_{1}=links[Alpha _{1},bright], ICH_{2}=links[Alpha _{2},Alpha _{1}Rechts],Punkte ,ICH_{n}=links[Alpha _{n},Alpha _{n-1}Rechts],Punkte } wo {Anzeigestil links(Alpha _{n}Rechts)_{n=1}^{unendlich }} is a strictly decreasing sequence such that {Anzeigestil Alpha _{n}rightarrow a.} In a similar way if {Anzeigestil I=[a,b),{Text{ mit }}bleq +infty } or if {Anzeigestil I=(a,b){Text{ mit }}-infty leq a 0 for each n. Definieren {Anzeigestil f_{n}(x)=0,,} zum {Anzeigestil ,,xx_{n}.} Then the jump function, or saltus-function, definiert von {Anzeigestil f(x)=,,Summe _{n=1}^{unendlich }f_{n}(x)=,,Summe _{x_{n}leq x}Lambda _{n}+Summe _{x_{n}

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