Discontinuities of monotone functions

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

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

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 ${\displaystyle f:I\to \mathbb {R} }$ 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

For a monotone function ${\displaystyle f}$, let ${\displaystyle f\nearrow }$ mean that ${\displaystyle f}$ is monotonically non-decreasing and let ${\displaystyle f\swarrow }$ mean that ${\displaystyle f}$ is monotonically non-increasing. 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

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

Examples. Let x₁ < x₂ < x₃ < ⋅⋅⋅ 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 (1990), 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 Im; that allows the total length to be computed ℓ(U)= Σ ℓ(Im). 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 Uc(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 Uc(f) is open and has total length ℓ(Uc(f)) ≤ 4 c^−1 (f(b) – f(a)). Note that Uc(f) consists the points x where the slope of h is greater that c near x. By definition Uc(f) is an open subset of (a, b), so can be written as a disjoint union of at most countably many open intervals Ik = (ak, bk). Let Jk be an interval with closure in Ik and ℓ(Jk) = ℓ(Ik)/2. By compactness, there are finitely many open intervals of the form (s,t) covering the closure of Jk. 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 (sk,1,tk,1), (sk,2,tk,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 Df(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 fn, write f = g + h with g = Σn≤N fn and h = Σn>N fn where N ≥ 1. Thus g is a step function having only finitely many discontinuities at xn for n ≤ N and h is a non-negative jump function. It follows that Df = g' +Dh = Dh except at the N points of discontinuity of g. Choosing N sufficiently large so that Σn>N λn + μn < 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

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

Notes

1. So for instance, these intervals need not be pairwise disjoint nor is it required that they intersect only at endpoints. It is even possible that ${\displaystyle \left[a_{n},b_{n}\right]\subseteq \left[a_{n+1},b_{n+1}\right]}$ for all ${\displaystyle n}$

References

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. 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. Apostol 1957.
9. Riesz & Sz.-Nagy 1990.
10. 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
    • Rubel 1963
    • Komornik 2016
15. Burkill 1951, pp. 10−11.
16. Rubel 1963
17. Komornik 2016
18. This is a simple example of how Lebesgue covering dimension applies in one real dimension; see for example Edgar (2008).

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