Discontinuities of monotone functions

Denote the limit from the left by

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

and denote the limit from the right by

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

If $f\backslash left(x^+\backslash right)$ and $f\backslash left(x^-\backslash right)$ exist and are finite then the difference $f\backslash left(x^+\backslash right)\; -\; f\backslash left(x^-\backslash right)$ is called the jump{{sfn|Nicolescu|Dinculeanu|Marcus|1971|p=213}} of $f$ at $x.$

Consider a real-valued function $f$ of real variable $x$ defined in a neighborhood of a point $x.$ If $f$ is discontinuous at the point $x$ then the discontinuity will be a removable discontinuity, or an essential discontinuity, or a jump discontinuity (also called a discontinuity of the first kind).{{sfn|Rudin|1964|loc=Def. 4.26, pp. 81–82}}

If the function is continuous at $x$ then the jump at $x$ is zero. Moreover, if $f$ is not continuous at $x,$ the jump can be zero at $x$ if $f\backslash left(x^+\backslash right)\; =\; f\backslash left(x^-\backslash right)\; \backslash neq\; f(x).$

Let $f$ be a real-valued monotone function defined on an interval $I.$ Then the set of discontinuities of the first kind is at most countable.

One can prove{{sfn|Rudin|1964|loc=Corollary, p. 83}}{{sfn|Nicolescu|Dinculeanu|Marcus|1971|p=213}} 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 $f$ be a monotone function defined on an interval $I.$ Then the set of discontinuities is at most countable.

This proof starts by proving the special case where the function's domain is a closed and bounded interval $[a,\; b].${{sfn|Apostol|1957|pp=162–3}}{{sfn|Hobson|1907|p=245}} The proof of the general case follows from this special case.

Two proofs of this special case are given.

Let $I\; :=\; [a,\; b]$ be an interval and let $f\; :\; I\; \backslash to\; \backslash R$ be a non-decreasing function (such as an increasing function).

Then for any

$$f(a)\; ~\backslash leq~\; f\backslash left(a^+\backslash right)\; ~\backslash leq~\; f\backslash left(x^-\backslash right)\; ~\backslash leq~\; f\backslash left(x^+\backslash right)\; ~\backslash leq~\; f\backslash left(b^-\backslash right)\; ~\backslash leq~\; f(b).$$

Let $\backslash alpha\; >\; 0$ and let be $n$ points inside $I$ at which the jump of $f$ is greater or equal to $\backslash alpha$:

$$f\backslash left(x\_i^+\backslash right)\; -\; f\backslash left(x\_i^-\backslash right)\; \backslash geq\; \backslash alpha,\backslash \; i=1,2,\backslash ldots,n$$

For any $i=1,2,\backslash ldots,n,$ $f\backslash left(x\_i^+\backslash right)\; \backslash leq\; f\backslash left(x\_\{i+1\}^-\backslash right)$ so that $f\backslash left(x\_\{i+1\}^-\backslash right)\; -\; f\backslash left(x\_i^+\backslash right)\; \backslash geq\; 0.$

Consequently,

$$\backslash begin\{alignat\}\{9\}$$

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{alignat}

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

Since we have that the number of points at which the jump is greater than $\backslash alpha$ is finite (possibly even zero).

Define the following sets:

$$S\_1:\; =\; \backslash left\backslash \{x\; :\; x\; \backslash in\; I,\; f\backslash left(x^+\backslash right)\; -\; f\backslash left(x^-\backslash right)\; \backslash geq\; 1\backslash right\backslash \},$$

Each set $S\_n$ is finite or the empty set. The union

$S\; =\; \backslash bigcup\_\{n=1\}^\backslash infty\; S\_n$ contains all points at which the jump is positive and hence contains all points of discontinuity. Since every $S\_i,\backslash \; i=1,2,\backslash ldots$ is at most countable, their union $S$ is also at most countable.

If $f$ 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.

$\backslash blacksquare$

For a monotone function $f$, let $f\backslash nearrow$ mean that $f$ is monotonically non-decreasing and let $f\backslash swarrow$ mean that $f$ is monotonically non-increasing. Let $f\; :\; [a,\; b]\; \backslash to\; \backslash R$ is a monotone function and let $D$ denote the set of all points $d\; \backslash in\; [a,\; b]$ in the domain of $f$ at which $f$ is discontinuous (which is necessarily a jump discontinuity).

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

It will now be shown that if $d,\; e\; \backslash in\; D$ are distinct, say with then $y\_d\; \backslash neq\; y\_e.$

If $f\; \backslash nearrow$ then implies $f\backslash left(d^+\backslash right)\; \backslash leq\; f\backslash left(e^-\backslash right)$ so that

If on the other hand $f\; \backslash searrow$ then implies $f\backslash left(d^+\backslash right)\; \backslash geq\; f\backslash left(e^-\backslash right)$ so that $y\_d\; >\; f\backslash left(d^+\backslash right)\; \backslash geq\; f\backslash left(e^-\backslash right)\; >\; y\_e.$

Either way, $y\_d\; \backslash neq\; y\_e.$

Thus every $d\; \backslash in\; D$ is associated with a unique rational number (said differently, the map $D\; \backslash to\; \backslash Q$ defined by $d\; \backslash mapsto\; y\_d$ is injective).

Since $\backslash Q$ is countable, the same must be true of $D.$

$\backslash blacksquare$

Suppose that the domain of $f$ (a monotone real-valued function) is equal to a union of countably many closed and bounded intervals; say its domain is $\backslash bigcup\_\{n\}\; \backslash left[a\_n,\; b\_n\backslash right]$ (no requirements are placed on these closed and bounded intervals{{efn|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 $\backslash left[a\_n,\; b\_n\backslash right]\; \backslash subseteq\; \backslash left[a\_\{n+1\},\; b\_\{n+1\}\backslash right]$ for all $n$}}).

It follows from the special case proved above that for every index $n,$ the restriction $f\backslash big\backslash vert\_\{\backslash left[a\_n,\; b\_n\backslash right]\}\; :\; \backslash left[a\_n,\; b\_n\backslash right]\; \backslash to\; \backslash R$ of $f$ to the interval $\backslash left[a\_n,\; b\_n\backslash right]$ has at most countably many discontinuities; denote this (countable) set of discontinuities by $D\_n.$

If $f$ has a discontinuity at a point $x\_0\; \backslash in\; \backslash bigcup\_\{n\}\; \backslash left[a\_n,\; b\_n\backslash right]$ in its domain then either $x\_0$ is equal to an endpoint of one of these intervals (that is, $x\_0\; \backslash in\; \backslash left\backslash \{a\_1,\; b\_1,\; a\_2,\; b\_2,\; \backslash ldots\backslash right\backslash \}$) or else there exists some index $n$ such that in which case $x\_0$ must be a point of discontinuity for $f\backslash big\backslash vert\_\{\backslash left[a\_n,\; b\_n\backslash right]\}$ (that is, $x\_0\; \backslash in\; D\_n$).

Thus the set $D$ of all points of at which $f$ is discontinuous is a subset of $\backslash left\backslash \{a\_1,\; b\_1,\; a\_2,\; b\_2,\; \backslash ldots\backslash right\backslash \}\; \backslash cup\; \backslash bigcup\_\{n\}\; D\_n,$ which is a countable set (because it is a union of countably many countable sets) so that its subset $D$ 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 $f$ is an interval $I$ 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 $I\_n$ with the property that any two consecutive intervals have an endpoint in common: $I\; =\; \backslash cup\_\{n=1\}^\backslash infty\; I\_n.$

If $I\; =\; (a,b]\; \backslash text\{\; with\; \}\; a\; \backslash geq\; -\backslash infty$ then $I\_1\; =\; \backslash left[\backslash alpha\_1,\; b\backslash right],\backslash \; I\_2\; =\; \backslash left[\backslash alpha\_2,\; \backslash alpha\_1\backslash right],\; \backslash ldots,\; I\_n\; =\; \backslash left[\backslash alpha\_n,\; \backslash alpha\_\{n-1\}\backslash right],\; \backslash ldots$ where $\backslash left(\backslash alpha\_n\backslash right)\_\{n=1\}^\{\backslash infty\}$ is a strictly decreasing sequence such that $\backslash alpha\_n\; \backslash rightarrow\; a.$ In a similar way if $I\; =\; [a,b),\; \backslash text\{\; with\; \}\; b\; \backslash leq\; +\backslash infty$ or if

In any interval $I\_n,$ 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.

$\backslash blacksquare$

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

:$f(x)\; =\; \backslash sum\_\{n=1\}^\{\backslash infty\}\; \backslash mu\_n\; \backslash chi\_\{[x\_n,b]\}\; (x)$

where χ_A denotes the characteristic function of a compact interval {{mvar|A}}. Then {{mvar|f}} is a non-decreasing function on [{{mvar|a}},{{mvar|b}}], which is continuous except for jump discontinuities at {{mvar|x}}_{{mvar|n}} for {{mvar|n}} ≥ 1. In the case of finitely many jump discontinuities, {{mvar|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.{{sfn|Apostol|1957}}{{sfn|Riesz|Sz.-Nagy|1990}}

More generally, the analysis of monotone functions has been studied by many mathematicians, starting from Abel, Jordan and Darboux. Following {{harvtxt|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 [{{mvar|a}},{{mvar|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 {{mvar|x_n}} ({{mvar|n}} ≥ 1) lie in ({{mvar|a}}, {{mvar|b}}) and take λ₁, λ₂, λ₃, ... and μ₁, μ₂, μ₃, ... non-negative with finite sum and with λ_{{mvar|n}} + μ_{{mvar|n}} > 0 for each {{mvar|n}}. Define

:$f\_n(x)=0\backslash ,\backslash ,$ for for $\backslash ,\backslash ,\; x\; >\; x\_n.$

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

:

is non-decreasing on [{{mvar|a}}, {{mvar|b}}] and is continuous except for jump discontinuities at {{mvar|x_n}} for {{mvar|n}} ≥ 1.{{sfn|Saks|1937}}{{sfn|Natanson|1955}}{{sfn|Łojasiewicz|1988}}

To prove this, note that sup |{{mvar|f}}_{{mvar|n}}| = λ_{{mvar|n}} + μ_{{mvar|n}}, so that Σ {{mvar|f}}_{{mvar|n}} converges uniformly to {{mvar|f}}. Passing to the limit, it follows that

:$f(x\_n)-f(x\_n-0)=\backslash lambda\_n,\backslash ,\backslash ,\backslash ,\; f(x\_n+0)-f(x\_n)=\backslash mu\_n,\backslash ,\backslash ,\backslash ,$ and $\backslash ,\backslash ,\; f(x\backslash pm\; 0)=f(x)$

if {{mvar|x}} is not one of the {{mvar|x}}_{{mvar|n}}'s.

Conversely, by a differentiation theorem of Lebesgue, the jump function {{mvar|f}} is uniquely determined by the properties:For more details, see

• {{harvnb|Riesz|Sz.-Nagy|1990}}
• {{harvnb|Young|Young|1911}}
• {{harvnb|von Neumann|1950}}
• {{harvnb|Boas|1961}}
• {{harvnb|Lipiński|1961}}
• {{harvnb|Rubel|1963}}
• {{harvnb|Komornik|2016}}

(1) being non-decreasing and non-positive; (2) having given jump data at its points of discontinuity {{mvar|x}}_{{mvar|n}}; (3) satisfying the boundary condition {{mvar|f}}({{mvar|a}}) = 0; and (4) having zero derivative almost everywhere.

{{Collapse top|title=Proof that a jump function has zero derivative almost everywhere.|left=yes}}

Property (4) can be checked following {{harvtxt|Riesz|Sz.-Nagy|1990}}, {{harvtxt|Rubel|1963}} and {{harvtxt|Komornik|2016}}. Without loss of generality, it can be assumed that {{mvar|f}} is a non-negative jump function defined on the compact [{{mvar|a}},{{mvar|b}}], with discontinuities only in ({{mvar|a}},{{mvar|b}}).

Note that an open set {{mvar|U}} of ({{mvar|a}},{{mvar|b}}) is canonically the disjoint union of at most countably many open intervals {{mvar|I}}_{{mvar|m}}; that allows the total length to be computed ℓ({{mvar|U}})= Σ ℓ({{mvar|I}}_{{mvar|m}}). Recall that a null set {{mvar|A}} is a subset such that, for any arbitrarily small ε' > 0, there is an open {{mvar|U}} containing {{mvar|A}} with ℓ({{mvar|U}}) < ε'. A crucial property of length is that, if {{mvar|U}} and {{mvar|V}} are open in ({{mvar|a}},{{mvar|b}}), then ℓ({{mvar|U}}) + ℓ({{mvar|V}}) = ℓ({{mvar|U}} ∪ {{mvar|V}}) + ℓ({{mvar|U}} ∩ {{mvar|V}}).{{sfn|Burkill|1951|pp=10−11}} It implies immediately that the union of two null sets is null; and that a finite or countable set is null.{{harvnb|Rubel|1963}}{{harvnb|Komornik|2016}}

Proposition 1. For {{mvar|c}} > 0 and a normalised non-negative jump function {{mvar|f}}, let {{mvar|U}}_{{mvar|c}}({{mvar|f}}) be the set of points {{mvar|x}} such that

:$\{f(t)-f(s)\backslash over\; t\; -s\}\; >\; c$

for some {{mvar|s}}, {{mvar|t}} with {{mvar|s}} < {{mvar|x}} < {{mvar|t}}. Then

{{mvar|U}}_{{mvar|c}}({{mvar|f}}) is open and has total length ℓ({{mvar|U}}_{{mvar|c}}({{mvar|f}})) ≤ 4 {{mvar|c}}⁻¹ ({{mvar|f}}({{mvar|b}}) – {{mvar|f}}({{mvar|a}})).

Note that {{mvar|U}}_{{mvar|c}}({{mvar|f}}) consists the points {{mvar|x}} where the slope of {{mvar|h}} is greater that {{mvar|c}} near {{mvar|x}}. By definition {{mvar|U}}_{{mvar|c}}({{mvar|f}}) is an open subset of ({{mvar|a}}, {{mvar|b}}), so can be written as a disjoint union of at most countably many open intervals {{mvar|I}}_{{mvar|k}} = ({{mvar|a}}_{{mvar|k}}, {{mvar|b}}_{{mvar|k}}). Let {{mvar|J}}_{{mvar|k}} be an interval with closure in {{mvar|I}}_{{mvar|k}} and ℓ({{mvar|J}}_{{mvar|k}}) = ℓ({{mvar|I}}_{{mvar|k}})/2. By compactness, there are finitely many open intervals of the form ({{mvar|s}},{{mvar|t}}) covering the closure of {{mvar|J}}_{{mvar|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 ({{mvar|s}}_{{mvar|k,1}},{{mvar|t}}_{{mvar|k,1}}), ({{mvar|s}}_{{mvar|k,2}},{{mvar|t}}_{{mvar|k,2}}), ... only intersecting at consecutive intervals.This is a simple example of how Lebesgue covering dimension applies in one real dimension; see for example {{harvtxt|Edgar|2008}}. Hence

:$\backslash ell(J\_k)\; \backslash le\; \backslash sum\_m\; (t\_\{k,m\}\; -\; s\_\{k,m\})\; \backslash le\; \backslash sum\_m\; c^\{-1\}(f(t\_\{k,m\})-f(s\_\{k,m\}))\; \backslash le\; 2\; c^\{-1\}(f(b\_k)-f(a\_k)).$

Finally sum both sides over {{mvar|k}}.

Proposition 2. If {{mvar|f}} is a jump function, then {{mvar|f}} '({{mvar|x}}) = 0 almost everywhere.

To prove this, define

:

a variant of the Dini derivative of {{mvar|f}}. It will suffice to prove that for any fixed {{mvar|c}} > 0, the Dini derivative satisfies {{mvar|D}}{{mvar|f}}({{mvar|x}}) ≤ {{mvar|c}} almost everywhere, i.e. on a null set.

Choose ε > 0, arbitrarily small. Starting from the definition of the jump function {{mvar|f}} = Σ {{mvar|f}}_{{mvar|n}}, write {{mvar|f}} = {{mvar|g}} + {{mvar|h}} with {{mvar|g}} = Σ_{{{mvar|n}}≤{{mvar|N}}} {{mvar|f}}_{{mvar|n}} and {{var|h}} = Σ_{{{mvar|n}}>{{mvar|N}}} {{mvar|f}}_{{mvar|n}} where {{mvar|N}} ≥ 1. Thus {{mvar|g}} is a step function having only finitely many discontinuities at {{mvar|x}}_{{mvar|n}} for {{mvar|n}} ≤ {{mvar|N}} and {{mvar|h}} is a non-negative jump function. It follows that {{mvar|D}}{{mvar|f}} = {{mvar|g}}' +{{mvar|D}}{{mvar|h}} = {{mvar|D}}{{mvar|h}} except at the {{mvar|N}} points of discontinuity of {{mvar|g}}. Choosing {{mvar|N}} sufficiently large so that Σ_{{{mvar|n}}>{{mvar|N}}} λ_{{mvar|n}} + μ_{{mvar|n}} < ε, it follows that {{mvar|h}} is a jump function such that {{mvar|h}}({{mvar|b}}) − {{mvar|h}}({{mvar|a}}) < ε and {{mvar|Dh}} ≤ {{mvar|c}} off an open set with length less than 4ε/{{mvar|c}}.

By construction {{mvar|Df}} ≤ {{mvar|c}} off an open set with length less than 4ε/{{mvar|c}}. Now set ε' = 4ε/{{mvar|c}} — then ε' and {{mvar|c}} are arbitrarily small and {{mvar|Df}} ≤ {{mvar|c}} off an open set of length less than ε'. Thus {{mvar|Df}} ≤ {{mvar|c}} almost everywhere. Since {{mvar|c}} could be taken arbitrarily small, {{mvar|Df}} and hence also {{mvar|f}} ' must vanish almost everywhere.

{{Collapse bottom}}

As explained in {{harvtxt|Riesz|Sz.-Nagy|1990}}, every non-decreasing non-negative function {{mvar|F}} can be decomposed uniquely as a sum of a jump function {{mvar|f}} and a continuous monotone function {{mvar|g}}: the jump function {{mvar|f}} is constructed by using the jump data of the original monotone function {{mvar|F}} and it is easy to check that {{mvar|g}} = {{mvar|F}} − {{mvar|f}} is continuous and monotone.{{harvnb|Riesz|Sz.-Nagy|1990|pp=13–15}}

• {{annotated link|Continuous function}}
• {{annotated link|Bounded variation}}
• Monotone function

{{notelist}}

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| year = 1964}}

• {{cite book|mr=0032011|last=von Neumann|first=John|author-link = John von Neumann|title=Functional Operators. I. Measures and Integrals|series=Annals of Mathematics Studies|volume= 21|publisher= Princeton University Press|year=1950

|doi=10.1515/9781400881895|chapter= IX. Monotonic Functions|pages=63–82|isbn=978-1-4008-8189-5}}

• {{cite journal|url=https://londmathsoc.onlinelibrary.wiley.com/doi/epdf/10.1112/plms/s2-9.1.325|first1=William Henry|last1=Young|first2=Grace Chisholm|last2=Young|journal=Proc. London Math. Soc.|title=On the Existence of a Differential Coefficient|year=1911|pages=325–335

|series=2|volume=9|issue=1|doi=10.1112/plms/s2-9.1.325}}

Category:Articles containing proofs

Category:Theory of continuous functions

Category:Theorems in real analysis