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