# Discontinuities of monotone functions 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. Contents 1 Definitions 2 Precise statement 3 Proofs 3.1 Proof when the domain is closed and bounded 3.1.1 Proof 1 3.1.2 Proof 2 3.2 Proof of general case 4 Jump functions 5 See also 6 Notes 7 References 8 Bibliography Definitions Denote the limit from the left by {displaystyle fleft(x^{-}right):=lim _{znearrow x}f(z)=lim _{stackrel {hto 0}{h>0}}f(x-h)} and denote the limit from the right by {displaystyle fleft(x^{+}right):=lim _{zsearrow x}f(z)=lim _{stackrel {hto 0}{h>0}}f(x+h).} If {displaystyle fleft(x^{+}right)} and {displaystyle fleft(x^{-}right)} exist and are finite then the difference {displaystyle fleft(x^{+}right)-fleft(x^{-}right)} is called the jump of {displaystyle f} at {displaystyle x.} Consider a real-valued function {displaystyle f} of real variable {displaystyle x} defined in a neighborhood of a point {displaystyle x.} If {displaystyle f} is discontinuous at the point {displaystyle 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 {displaystyle x} then the jump at {displaystyle x} is zero. Moreover, if {displaystyle f} is not continuous at {displaystyle x,} the jump can be zero at {displaystyle x} if {displaystyle fleft(x^{+}right)=fleft(x^{-}right)neq f(x).} Precise statement Let {displaystyle 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, of the first kind. With this remark the theorem takes the stronger form: Let {displaystyle 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 {displaystyle [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.

Proof 1 Let {displaystyle I:=[a,b]} be an interval and let {displaystyle f:Ito mathbb {R} } be a non-decreasing function (such as an increasing function). Then for any {displaystyle a0} and let {displaystyle x_{1}y_{d}>fleft(d^{+}right)} holds).

It will now be shown that if {displaystyle d,ein D} are distinct, say with {displaystyle dfleft(d^{+}right)geq fleft(e^{-}right)>y_{e}.} Either way, {displaystyle 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} } defined by {displaystyle dmapsto y_{d}} is injective). Since {displaystyle mathbb {Q} } is countable, the same must be true of {displaystyle D.} {displaystyle blacksquare } Proof of general case Suppose that the domain of {displaystyle 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}left[a_{n},b_{n}right]} (no requirements are placed on these closed and bounded intervals[a]). It follows from the special case proved above that for every index {displaystyle n,} the restriction {displaystyle f{big vert }_{left[a_{n},b_{n}right]}:left[a_{n},b_{n}right]to mathbb {R} } of {displaystyle f} to the interval {displaystyle left[a_{n},b_{n}right]} has at most countably many discontinuities; denote this (countable) set of discontinuities by {displaystyle D_{n}.} If {displaystyle f} has a discontinuity at a point {displaystyle x_{0}in bigcup _{n}left[a_{n},b_{n}right]} in its domain then either {displaystyle x_{0}} is equal to an endpoint of one of these intervals (that is, {displaystyle x_{0}in left{a_{1},b_{1},a_{2},b_{2},ldots right}} ) or else there exists some index {displaystyle n} such that {displaystyle a_{n} 0 for each n. Define {displaystyle f_{n}(x)=0,,} for {displaystyle ,,xx_{n}.} Then the jump function, or saltus-function, defined by {displaystyle f(x)=,,sum _{n=1}^{infty }f_{n}(x)=,,sum _{x_{n}leq x}lambda _{n}+sum _{x_{n}

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