Monotone and Right Continuous Has Finitely Many Discontinuities

Monotone maps have countable discontinuities

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]

Definitions [edit]

Denote the limit from the left by

$${\displaystyle f\left(x^{-}\right):=\lim _{z\nearrow x}f(z)=\lim _{\stackrel {h\to 0}{h>0}}f(x-h)}$$

and denote the limit from the right by

$${\displaystyle f\left(x^{+}\right):=\lim _{z\searrow x}f(z)=\lim _{\stackrel {h\to 0}{h>0}}f(x+h).}$$

If ${\displaystyle f\left(x^{+}\right)}$ and ${\displaystyle f\left(x^{-}\right)}$ exist and are finite then the difference ${\displaystyle f\left(x^{+}\right)-f\left(x^{-}\right)}$ 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 ${\displaystyle f\left(x^{+}\right)=f\left(x^{-}\right)\neq f(x).}$

Precise statement [edit]

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 [edit]

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 [edit]

Two proofs of this special case are given.

Proof 1 [edit]

Let ${\displaystyle I:=[a,b]}$ be an interval and let ${\displaystyle f:I\to \mathbb {R} }$ be a non-decreasing function (such as an increasing function). Then for any ${\displaystyle a<x<b,}$

$${\displaystyle f(a)~\leq ~f\left(a^{+}\right)~\leq ~f\left(x^{-}\right)~\leq ~f\left(x^{+}\right)~\leq ~f\left(b^{-}\right)~\leq ~f(b).}$$

Let ${\displaystyle \alpha >0}$ and let ${\displaystyle x_{1}<x_{2}<\cdots <x_{n}}$ be ${\displaystyle n}$ points inside ${\displaystyle I}$ at which the jump of ${\displaystyle f}$ is greater or equal to ${\displaystyle \alpha }$ :

$${\displaystyle f\left(x_{i}^{+}\right)-f\left(x_{i}^{-}\right)\geq \alpha ,\ i=1,2,\ldots ,n}$$

For any ${\displaystyle i=1,2,\ldots ,n,}$ ${\displaystyle f\left(x_{i}^{+}\right)\leq f\left(x_{i+1}^{-}\right)}$ so that ${\displaystyle f\left(x_{i+1}^{-}\right)-f\left(x_{i}^{+}\right)\geq 0.}$ Consequently,

$${\displaystyle {\begin{alignedat}{9}f(b)-f(a)&\geq f\left(x_{n}^{+}\right)-f\left(x_{1}^{-}\right)\\&=\sum _{i=1}^{n}\left[f\left(x_{i}^{+}\right)-f\left(x_{i}^{-}\right)\right]+\sum _{i=1}^{n-1}\left[f\left(x_{i+1}^{-}\right)-f\left(x_{i}^{+}\right)\right]\\&\geq \sum _{i=1}^{n}\left[f\left(x_{i}^{+}\right)-f\left(x_{i}^{-}\right)\right]\\&\geq n\alpha \end{alignedat}}}$$

and hence ${\displaystyle n\leq {\frac {f(b)-f(a)}{\alpha }}.}$

Since ${\displaystyle f(b)-f(a)<\infty }$ we have that the number of points at which the jump is greater than ${\displaystyle \alpha }$ is finite (possibly even zero).

Define the following sets:

$${\displaystyle S_{1}:=\left\{x:x\in I,f\left(x^{+}\right)-f\left(x^{-}\right)\geq 1\right\},}$$

$${\displaystyle S_{n}:=\left\{x:x\in I,{\frac {1}{n}}\leq f\left(x^{+}\right)-f\left(x^{-}\right)<{\frac {1}{n-1}}\right\},\ n\geq 2.}$$

Each set ${\displaystyle S_{n}}$ is finite or the empty set. The union ${\displaystyle S=\bigcup _{n=1}^{\infty }S_{n}}$ contains all points at which the jump is positive and hence contains all points of discontinuity. Since every ${\displaystyle S_{i},\ i=1,2,\ldots }$ is at most countable, their union ${\displaystyle S}$ is also at most countable.

If ${\displaystyle f}$ is non-increasing (or decreasing) then the proof is similar. This completes the proof of the special case where the function's domain is a closed and bounded interval. ${\displaystyle \blacksquare }$

Proof 2 [edit]

So let ${\displaystyle f:[a,b]\to \mathbb {R} }$ is a monotone function and let ${\displaystyle D}$ denote the set of all points ${\displaystyle d\in [a,b]}$ in the domain of ${\displaystyle f}$ at which ${\displaystyle f}$ is discontinuous (which is necessarily a jump discontinuity).

Because ${\displaystyle f}$ has a jump discontinuity at ${\displaystyle d\in D,}$ ${\displaystyle f\left(d^{-}\right)\neq f\left(d^{+}\right)}$ so there exists some rational number ${\displaystyle y_{d}\in \mathbb {Q} }$ that lies strictly in between ${\displaystyle f\left(d^{-}\right){\text{ and }}f\left(d^{+}\right)}$ (specifically, if ${\displaystyle f\nearrow }$ then pick ${\displaystyle y_{d}\in \mathbb {Q} }$ so that ${\displaystyle f\left(d^{-}\right)<y_{d}<f\left(d^{+}\right)}$ while if ${\displaystyle f\searrow }$ then pick ${\displaystyle y_{d}\in \mathbb {Q} }$ so that ${\displaystyle f\left(d^{-}\right)>y_{d}>f\left(d^{+}\right)}$ holds).

It will now be shown that if ${\displaystyle d,e\in D}$ are distinct, say with ${\displaystyle d<e,}$ then ${\displaystyle y_{d}\neq y_{e}.}$ If ${\displaystyle f\nearrow }$ then ${\displaystyle d<e}$ implies ${\displaystyle f\left(d^{+}\right)\leq f\left(e^{-}\right)}$ so that ${\displaystyle y_{d}<f\left(d^{+}\right)\leq f\left(e^{-}\right)<y_{e}.}$ If on the other hand ${\displaystyle f\searrow }$ then ${\displaystyle d<e}$ implies ${\displaystyle f\left(d^{+}\right)\geq f\left(e^{-}\right)}$ so that ${\displaystyle y_{d}>f\left(d^{+}\right)\geq f\left(e^{-}\right)>y_{e}.}$ Either way, ${\displaystyle y_{d}\neq y_{e}.}$

Thus every ${\displaystyle d\in D}$ is associated with a unique rational number (said differently, the map ${\displaystyle D\to \mathbb {Q} }$ defined by ${\displaystyle d\mapsto y_{d}}$ is injective). Since ${\displaystyle \mathbb {Q} }$ is countable, the same must be true of ${\displaystyle D.}$ ${\displaystyle \blacksquare }$

Proof of general case [edit]

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 ${\displaystyle a_{n}<x_{0}<b_{n},}$ in which case ${\displaystyle x_{0}}$ must be a point of discontinuity for ${\displaystyle f{\big \vert }_{\left[a_{n},b_{n}\right]}}$ (that is, ${\displaystyle x_{0}\in D_{n}}$ ). Thus the set ${\displaystyle D}$ of all points of at which ${\displaystyle f}$ is discontinuous is a subset of ${\displaystyle \left\{a_{1},b_{1},a_{2},b_{2},\ldots \right\}\cup \bigcup _{n}D_{n},}$ which is a countable set (because it is a union of countably many countable sets) so that its subset ${\displaystyle D}$ must also be countable (because every subset of a countable set is countable).

In particular, 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 ${\displaystyle f}$ is an interval ${\displaystyle I}$ 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 ${\displaystyle I_{n}}$ with the property that any two consecutive intervals have an endpoint in common: ${\displaystyle I=\cup _{n=1}^{\infty }I_{n}.}$ If ${\displaystyle I=(a,b]{\text{ with }}a\geq -\infty }$ then ${\displaystyle I_{1}=\left[\alpha _{1},b\right],\ I_{2}=\left[\alpha _{2},\alpha _{1}\right],\ldots ,I_{n}=\left[\alpha _{n},\alpha _{n-1}\right],\ldots }$ where ${\displaystyle \left(\alpha _{n}\right)_{n=1}^{\infty }}$ is a strictly decreasing sequence such that ${\displaystyle \alpha _{n}\rightarrow a.}$ In a similar way if ${\displaystyle I=[a,b),{\text{ with }}b\leq +\infty }$ or if ${\displaystyle I=(a,b){\text{ with }}-\infty \leq a<b\leq \infty .}$ In any interval ${\displaystyle I_{n},}$ there are at most countable many points of discontinuity, and since a countable union of at most countable sets is at most countable, it follows that the set of all discontinuities is at most countable. ${\displaystyle \blacksquare }$

Jump functions [edit]

Examples. Let x ₁ < x ₂ < x ₃ < ⋅⋅⋅ be a countable subset of the compact interval [ a , b ] and let μ₁, μ₂, μ₃, ... be a positive sequence with finite sum. Set

${\displaystyle f(x)=\sum _{n=1}^{\infty }\mu _{n}\chi _{[x_{n},b]}(x)}$

where χ _A denotes the characteristic function of a compact interval A . Then f is a non-decreasing function on [ a , b ], which is continuous except for jump discontinuities at x _n for n ≥ 1. In the case of finitely many jump discontinuities, f is a step function. The examples above are generalised step functions; they are very special cases of what are called jump functions or saltus-functions.^{[8] [9]}

More generally, the analysis of monotone functions has been studied by many mathematicians, starting from Abel, Jordan and Darboux. Following Riesz & Sz.-Nagy (1990), replacing a function by its negative if necessary, only the case of non-negative non-decreasing functions has to be considered. The domain [ a , b ] can be finite or have ∞ or −∞ as endpoints.

The main task is to construct monotone functions — generalising step functions — with discontinuities at a given denumerable set of points and with prescribed left and right discontinuities at each of these points. Let x _n ( n ≥ 1) lie in ( a , b ) and take λ₁, λ₂, λ₃, ... and μ₁, μ₂, μ₃, ... non-negative with finite sum and with λ _n + μ _n > 0 for each n . Define

${\displaystyle f_{n}(x)=0\,\,}$ for ${\displaystyle \,\,x<x_{n},\,\,f_{n}(x_{n})=\lambda _{n},\,\,f_{n}(x)=\lambda _{n}+\mu _{n}\,\,}$ for ${\displaystyle \,\,x>x_{n}.}$

Then the jump function, or saltus-function, defined by

${\displaystyle f(x)=\,\,\sum _{n=1}^{\infty }f_{n}(x)=\,\,\sum _{x_{n}\leq x}\lambda _{n}+\sum _{x_{n}<x}\mu _{n},}$

is non-decreasing on [ a , b ] and is continuous except for jump discontinuities at x _n for n ≥ 1.^{[10] [11] [12] [13]}

To prove this, note that sup | f _n | = λ _n + μ _n , so that Σ f _n converges uniformly to f . Passing to the limit, it follows that

${\displaystyle f(x_{n})-f(x_{n}-0)=\lambda _{n},\,\,\,f(x_{n}+0)-f(x_{n})=\mu _{n},\,\,\,}$ and ${\displaystyle \,\,f(x\pm 0)=f(x)}$

if x is not one of the x _n 's.^[10]

Conversely, by a differentiation theorem of Lebesgue, the jump function f is uniquely determined by the properties:^[14] (1) being non-decreasing and non-positive; (2) having given jump data at its points of discontinuity x _n ; (3) satisfying the boundary condition f ( a ) = 0; and (4) having zero derivative almost everywhere.

Proof that a jump function has zero derivative almost everywhere.

Property (4) can be checked following Riesz & Sz.-Nagy (1955) harvtxt error: no target: CITEREFRieszSz.-Nagy1955 (help), Rubel (1963) and Komornik (2016). Without loss of generality, it can be assumed that f is a non-negative jump function defined on the compact [ a , b ], with discontinuities only in ( a , b ). Note that an open set U of ( a , b ) is canonically the disjoint union of at most countably many open intervals I m ; that allows the total length to be computed ℓ( U )= Σ ℓ( I m ). Recall that a null set A is a subset such that, for any arbitrarily small ε' > 0, there is an open U containing A with ℓ( U ) < ε'. A crucial property of length is that, if U and V are open in ( a , b ), then ℓ( U ) + ℓ( V ) = ℓ( U ∪ V ) + ℓ( U ∩ V ).[15] It implies immediately that the union of two null sets is null; and that a finite or countable set is null.[16] [17] Proposition 1. For c > 0 and a normalised non-negative jump function f , let U c ( f ) be the set of points x such that ${\displaystyle {f(t)-f(s) \over t-s}>c}$ for some s , t with s < x < t . Then U c ( f ) is open and has total length ℓ( U c ( f )) ≤ 4 c –1 ( f ( b ) – f ( a )). Note that U c ( f ) consists the points x where the slope of h is greater that c near x . By definition U c ( f ) is an open subset of ( a , b ), so can be written as a disjoint union of at most countably many open intervals I k = ( a k , b k ). Let J k be an interval with closure in I k and ℓ( J k ) = ℓ( I k )/2. By compactness, there are finitely many open intervals of the form ( s , t ) covering the closure of J k . On the other hand, it is elementary that, if three fixed bounded open intervals have a common point of intersection, then their union contains one of the three intervals: indeed just take the supremum and infimum points to identify the endpoints. As a result, the finite cover can be taken as adjacent open intervals ( s k,1 , t k,1 ), ( s k,2 , t k,2 ), ... only intersecting at consecutive intervals.[18] Hence ${\displaystyle \ell (J_{k})\leq \sum _{m}(t_{k,m}-s_{k,m})\leq \sum _{m}c^{-1}(f(t_{k,m})-f(s_{k,m}))\leq 2c^{-1}(f(b_{k})-f(a_{k})).}$ Finally sum both sides over k .[16] [17] Proposition 2. If f is a jump function, then f '( x ) = 0 almost everywhere. To prove this, define ${\displaystyle Df(x)=\limsup _{s,t\rightarrow x,\,\,s<x<t}{f(t)-f(s) \over t-s},}$ a variant of the Dini derivative of f . It will suffice to prove that for any fixed c > 0, the Dini derivative satisfies D f ( x ) ≤ c almost everywhere, i.e. on a null set. Choose ε > 0, arbitrarily small. Starting from the definition of the jump function f = Σ f n , write f = g + h with g = Σ n ≤ N f n and h = Σ n > N f n where N ≥ 1. Thus g is a step function having only finitely many discontinuities at x n for n ≤ N and h is a non-negative jump function. It follows that D f = g ' + D h = D h except at the N points of discontinuity of g . Choosing N sufficiently large so that Σ n > N λ n + μ n < ε, it follows that h is a jump function such that h ( b ) − h ( a ) < ε and Dh ≤ c off an open set with length less than 4ε/ c . By construction Df ≤ c off an open set with length less than 4ε/ c . Now set ε' = 4ε/ c — then ε' and c are arbitrarily small and Df ≤ c off an open set of length less than ε'. Thus Df ≤ c almost everywhere. Since c could be taken arbitrarily small, Df and hence also f ' must vanish almost everywhere.[16] [17]

As explained in Riesz & Sz.-Nagy (1990), every non-decreasing non-negative function F can be decomposed uniquely as a sum of a jump function f and a continuous monotone function g : the jump function f is constructed by using the jump data of the original monotone function F and it is easy to check that g = F − f is continuous and monotone.^[10]

See also [edit]

• Continuous function – Mathematical function with no sudden changes
• Bounded variation – Real function with finite total variation
• Monotone function

Notes [edit]

References [edit]

1. ^ Froda, Alexandre (3 December 1929). Sur la distribution des propriétés de voisinage des functions de variables réelles (PDF) (Thesis). Paris: Hermann. JFM 55.0742.02.
2. ^ Jean Gaston Darboux, Mémoire sur les fonctions discontinues, Annales Scientifiques de l'École Normale Supérieure, 2-ème série, t. IV, 1875, Chap VI.
3. ^ ^{a b} Nicolescu, Dinculeanu & Marcus 1971, p. 213.
4. ^ Rudin 1964, Def. 4.26, pp. 81–82.
5. ^ Rudin 1964, Corollary, p. 83.
6. ^ Apostol 1957, pp. 162–3.
7. ^ Hobson 1907, p. 245.
8. ^ Apsotol 1957. sfn error: no target: CITEREFApsotol1957 (help)
9. ^ Riesz & Sz.-Nagy 1990.
10. ^ ^{a b c} Riesz & Sz.-Nagy 1990, pp. 13–15
11. ^ Saks 1937.
12. ^ Natanson 1955.
13. ^ Łojasiewicz 1988.
14. ^ For more details, see
    • Riesz & Sz.-Nagy 1990
    • Young & Young 1911
    • von Neumann 1950
    • Boas 1961
    • Lipiński 1961
    • Lee 1963 harvnb error: no target: CITEREFLee1963 (help)
    • Komornik 2016
15. ^ Burkill 1951, pp. 10−11.
16. ^ ^{a b c} Rubel 1963
17. ^ ^{a b c} Komornik 2016
18. ^ This is a simple example of how Lebesgue covering dimension applies in one real dimension; see for example Edgar (2008).

Bibliography [edit]

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