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 (monotone) function are necessarily jump discontinuities and there are at most countably many of them.

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.[1] Prior work on discontinuities had already been discussed in the 1875 memoir of the French mathematician Jean Gaston Darboux.[2] 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[3] 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).[4] 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[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, 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].} [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.

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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