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Discontinuities of monotone functions
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<h2 id="definitions">Definitions</h2> <p>Denote the <a href="/facts/Limit_from_the_left/zLfr8gc5">limit from the left</a> by f ( x − ) := lim z ↗ x f ( z ) = lim h > 0 h → 0 f ( x − h ) {\displaystyle f\left(x^{-}\right):=\lim _{z\nearrow x}f(z)=\lim _{\stackrel {h\to 0}{h>0}}f(x-h)} and denote the <a href="/facts/Limit_from_the_right/zLfr8gc5">limit from the right</a> by f ( x + ) := lim z ↘ x f ( z ) = lim h > 0 h → 0 f ( x + h ) . {\displaystyle f\left(x^{+}\right):=\lim _{z\searrow x}f(z)=\lim _{\stackrel {h\to 0}{h>0}}f(x+h).} </p><p>If f ( x + ) {\displaystyle f\left(x^{+}\right)} and f ( x − ) {\displaystyle f\left(x^{-}\right)} exist and are finite then the difference f ( x + ) − f ( x − ) {\displaystyle f\left(x^{+}\right)-f\left(x^{-}\right)} is called the jump<a class="footnote-ref" id="fnref:3" href="#fn:3"><sup>3</sup></a> of f {\displaystyle f} at x . {\displaystyle x.} </p><p>Consider a real-valued function f {\displaystyle f} of real variable x {\displaystyle x} defined in a neighborhood of a point x . {\displaystyle x.} If f {\displaystyle f} is discontinuous at the point x {\displaystyle x} then the discontinuity will be a <a href="/facts/Removable_discontinuity/TGLMsQom">removable discontinuity</a>, or an <a href="/facts/Essential_discontinuity/TGLMsQom">essential discontinuity</a>, or a <a href="/facts/Jump_discontinuity/TGLMsQom">jump discontinuity</a> (also called a discontinuity of the first kind).<a class="footnote-ref" id="fnref:4" href="#fn:4"><sup>4</sup></a> If the function is continuous at x {\displaystyle x} then the jump at x {\displaystyle x} is zero. Moreover, if f {\displaystyle f} is not continuous at x , {\displaystyle x,} the jump can be zero at x {\displaystyle x} if f ( x + ) = f ( x − ) ≠ f ( x ) . {\displaystyle f\left(x^{+}\right)=f\left(x^{-}\right)\neq f(x).} </p> <h2 id="precise-statement">Precise statement</h2> <p>Let f {\displaystyle f} be a real-valued <a href="/facts/Monotonic_function/MjQK0YL2">monotone</a> function defined on an <a href="/facts/Interval_(mathematics)/e9se3vTe">interval</a> I . {\displaystyle I.} Then the set of discontinuities of the first kind is <a href="/facts/Countable_set/Wj4DmWPv">at most countable</a>. </p><p>One can prove<a class="footnote-ref" id="fnref:5" href="#fn:5"><sup>5</sup></a><a class="footnote-ref" id="fnref:6" href="#fn:6"><sup>6</sup></a> 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: </p><p>Let f {\displaystyle f} be a monotone function defined on an interval I . {\displaystyle I.} Then the set of discontinuities is at most countable. </p> <h2 id="proofs">Proofs</h2> <p>This proof starts by proving the special case where the function's domain is a closed and bounded interval [ a , b ] . {\displaystyle [a,b].} <a class="footnote-ref" id="fnref:7" href="#fn:7"><sup>7</sup></a><a class="footnote-ref" id="fnref:8" href="#fn:8"><sup>8</sup></a> The proof of the general case follows from this special case. </p> <h3>Proof when the domain is closed and bounded</h3> <p>Two proofs of this special case are given. </p> <h4>Proof 1</h4> <p>Let I := [ a , b ] {\displaystyle I:=[a,b]} be an interval and let f : I → R {\displaystyle f:I\to \mathbb {R} } be a non-decreasing function (such as an <a href="/facts/Monotonic_function/MjQK0YL2">increasing</a> function). Then for any a < x < b , {\displaystyle a<x<b,} f ( a ) ≤ f ( a + ) ≤ f ( x − ) ≤ f ( x + ) ≤ f ( b − ) ≤ f ( 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 α > 0 {\displaystyle \alpha >0} and let x 1 < x 2 < ⋯ < x n {\displaystyle x_{1}<x_{2}<\cdots <x_{n}} be n {\displaystyle n} points inside I {\displaystyle I} at which the jump of f {\displaystyle f} is greater or equal to α {\displaystyle \alpha } : f ( x i + ) − f ( x i − ) ≥ α , i = 1 , 2 , … , n {\displaystyle f\left(x_{i}^{+}\right)-f\left(x_{i}^{-}\right)\geq \alpha ,\ i=1,2,\ldots ,n} </p><p>For any i = 1 , 2 , … , n , {\displaystyle i=1,2,\ldots ,n,} f ( x i + ) ≤ f ( x i + 1 − ) {\displaystyle f\left(x_{i}^{+}\right)\leq f\left(x_{i+1}^{-}\right)} so that f ( x i + 1 − ) − f ( x i + ) ≥ 0. {\displaystyle f\left(x_{i+1}^{-}\right)-f\left(x_{i}^{+}\right)\geq 0.} Consequently, f ( b ) − f ( a ) ≥ f ( x n + ) − f ( x 1 − ) = ∑ i = 1 n [ f ( x i + ) − f ( x i − ) ] + ∑ i = 1 n − 1 [ f ( x i + 1 − ) − f ( x i + ) ] ≥ ∑ i = 1 n [ f ( x i + ) − f ( x i − ) ] ≥ n α {\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 n ≤ f ( b ) − f ( a ) α . {\displaystyle n\leq {\frac {f(b)-f(a)}{\alpha }}.} </p><p>Since f ( b ) − f ( a ) < ∞ {\displaystyle f(b)-f(a)<\infty } we have that the number of points at which the jump is greater than α {\displaystyle \alpha } is finite (possibly even zero). </p><p>Define the following sets: S 1 := { x : x ∈ I , f ( x + ) − f ( x − ) ≥ 1 } , {\displaystyle S_{1}:=\left\{x:x\in I,f\left(x^{+}\right)-f\left(x^{-}\right)\geq 1\right\},} S n := { x : x ∈ I , 1 n ≤ f ( x + ) − f ( x − ) < 1 n − 1 } , n ≥ 2. {\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.} </p><p>Each set S n {\displaystyle S_{n}} is finite or the <a href="/facts/Empty_set/ffEk3eA6">empty set</a>. The union S = ⋃ n = 1 ∞ S n {\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 S i , i = 1 , 2 , … {\displaystyle S_{i},\ i=1,2,\ldots } is at most countable, their union S {\displaystyle S} is also at most countable. </p><p>If f {\displaystyle f} is non-increasing (or <a href="/facts/Monotonic_function/MjQK0YL2">decreasing</a>) 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 } </p> <h4>Proof 2</h4> <p>For a monotone function f {\displaystyle f} , let f ↗ {\displaystyle f\nearrow } mean that f {\displaystyle f} is monotonically non-decreasing and let f ↙ {\displaystyle f\swarrow } mean that f {\displaystyle f} is monotonically non-increasing. Let f : [ a , b ] → R {\displaystyle f:[a,b]\to \mathbb {R} } is a monotone function and let D {\displaystyle D} denote the set of all points d ∈ [ a , b ] {\displaystyle d\in [a,b]} in the domain of f {\displaystyle f} at which f {\displaystyle f} is discontinuous (which is necessarily a jump discontinuity). </p><p>Because f {\displaystyle f} has a jump discontinuity at d ∈ D , {\displaystyle d\in D,} f ( d − ) ≠ f ( d + ) {\displaystyle f\left(d^{-}\right)\neq f\left(d^{+}\right)} so there exists some <a href="/facts/Rational_number/VJQmyY05">rational number</a> y d ∈ Q {\displaystyle y_{d}\in \mathbb {Q} } that lies strictly in between f ( d − ) and f ( d + ) {\displaystyle f\left(d^{-}\right){\text{ and }}f\left(d^{+}\right)} (specifically, if f ↗ {\displaystyle f\nearrow } then pick y d ∈ Q {\displaystyle y_{d}\in \mathbb {Q} } so that f ( d − ) < y d < f ( d + ) {\displaystyle f\left(d^{-}\right)<y_{d}<f\left(d^{+}\right)} while if f ↘ {\displaystyle f\searrow } then pick y d ∈ Q {\displaystyle y_{d}\in \mathbb {Q} } so that f ( d − ) > y d > f ( d + ) {\displaystyle f\left(d^{-}\right)>y_{d}>f\left(d^{+}\right)} holds). </p><p>It will now be shown that if d , e ∈ D {\displaystyle d,e\in D} are distinct, say with d < e , {\displaystyle d<e,} then y d ≠ y e . {\displaystyle y_{d}\neq y_{e}.} If f ↗ {\displaystyle f\nearrow } then d < e {\displaystyle d<e} implies f ( d + ) ≤ f ( e − ) {\displaystyle f\left(d^{+}\right)\leq f\left(e^{-}\right)} so that y d < f ( d + ) ≤ f ( e − ) < y e . {\displaystyle y_{d}<f\left(d^{+}\right)\leq f\left(e^{-}\right)<y_{e}.} If on the other hand f ↘ {\displaystyle f\searrow } then d < e {\displaystyle d<e} implies f ( d + ) ≥ f ( e − ) {\displaystyle f\left(d^{+}\right)\geq f\left(e^{-}\right)} so that y d > f ( d + ) ≥ f ( e − ) > y e . {\displaystyle y_{d}>f\left(d^{+}\right)\geq f\left(e^{-}\right)>y_{e}.} Either way, y d ≠ y e . {\displaystyle y_{d}\neq y_{e}.} </p><p>Thus every d ∈ D {\displaystyle d\in D} is associated with a unique rational number (said differently, the map D → Q {\displaystyle D\to \mathbb {Q} } defined by d ↦ y d {\displaystyle d\mapsto y_{d}} is <a href="/facts/Injective_function/5ES6tqf5">injective</a>). Since Q {\displaystyle \mathbb {Q} } is countable, the same must be true of D . {\displaystyle D.} ◼ {\displaystyle \blacksquare } </p> <h3>Proof of general case</h3> <p>Suppose that the domain of f {\displaystyle f} (a monotone real-valued function) is equal to a union of countably many closed and bounded intervals; say its domain is ⋃ n [ a n , b n ] {\displaystyle \bigcup _{n}\left[a_{n},b_{n}\right]} (no requirements are placed on these closed and bounded intervals<a class="footnote-ref" id="fnref:9" href="#fn:9"><sup>9</sup></a>). It follows from the special case proved above that for every index n , {\displaystyle n,} the <a href="/facts/Restriction_of_a_map/JV59DhKy">restriction</a> f | [ a n , b n ] : [ a n , b n ] → R {\displaystyle f{\big \vert }_{\left[a_{n},b_{n}\right]}:\left[a_{n},b_{n}\right]\to \mathbb {R} } of f {\displaystyle f} to the interval [ a n , b n ] {\displaystyle \left[a_{n},b_{n}\right]} has at most countably many discontinuities; denote this (countable) set of discontinuities by D n . {\displaystyle D_{n}.} If f {\displaystyle f} has a discontinuity at a point x 0 ∈ ⋃ n [ a n , b n ] {\displaystyle x_{0}\in \bigcup _{n}\left[a_{n},b_{n}\right]} in its domain then either x 0 {\displaystyle x_{0}} is equal to an endpoint of one of these intervals (that is, x 0 ∈ { a 1 , b 1 , a 2 , b 2 , … } {\displaystyle x_{0}\in \left\{a_{1},b_{1},a_{2},b_{2},\ldots \right\}} ) or else there exists some index n {\displaystyle n} such that a n < x 0 < b n , {\displaystyle a_{n}<x_{0}<b_{n},} in which case x 0 {\displaystyle x_{0}} must be a point of discontinuity for f | [ a n , b n ] {\displaystyle f{\big \vert }_{\left[a_{n},b_{n}\right]}} (that is, x 0 ∈ D n {\displaystyle x_{0}\in D_{n}} ). Thus the set D {\displaystyle D} of all points of at which f {\displaystyle f} is discontinuous is a subset of { a 1 , b 1 , a 2 , b 2 , … } ∪ ⋃ n D n , {\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 D {\displaystyle D} must also be countable (because every subset of a countable set is countable). </p><p>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. </p><p>To make this argument more concrete, suppose that the domain of f {\displaystyle f} is an interval I {\displaystyle I} that is not <a href="/facts/Closed_set/upYnRTUj">closed</a> and <a href="/facts/Bounded_set/yCldjtoy">bounded</a> (and hence by <a href="/facts/Heine%25E2%2580%2593Borel_theorem/c74qCFHY">Heine–Borel theorem</a> not <a href="/facts/Compact_set/d0cgXJH7">compact</a>). Then the interval can be written as a countable union of closed and bounded intervals I n {\displaystyle I_{n}} with the property that any two consecutive intervals have an <a href="/facts/Interval_(mathematics)/e9se3vTe">endpoint</a> in common: I = ∪ n = 1 ∞ I n . {\displaystyle I=\cup _{n=1}^{\infty }I_{n}.} If I = ( a , b ] with a ≥ − ∞ {\displaystyle I=(a,b]{\text{ with }}a\geq -\infty } then I 1 = [ α 1 , b ] , I 2 = [ α 2 , α 1 ] , … , I n = [ α n , α n − 1 ] , … {\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 ( α n ) n = 1 ∞ {\displaystyle \left(\alpha _{n}\right)_{n=1}^{\infty }} is a strictly decreasing <a href="/facts/Sequence/gV2S4uqp">sequence</a> such that α n → a . {\displaystyle \alpha _{n}\rightarrow a.} In a similar way if I = [ a , b ) , with b ≤ + ∞ {\displaystyle I=[a,b),{\text{ with }}b\leq +\infty } or if I = ( a , b ) with − ∞ ≤ a < b ≤ ∞ . {\displaystyle I=(a,b){\text{ with }}-\infty \leq a<b\leq \infty .} In any interval I n , {\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 } </p> <h2 id="jump-functions">Jump functions</h2> <p>Examples. Let <i>x</i>1 < <i>x</i>2 < <i>x</i>3 < ⋅⋅⋅ be a countable subset of the compact interval [<i>a</i>,<i>b</i>] and let μ1, μ2, μ3, ... be a positive sequence with finite sum. Set </p> f ( x ) = ∑ n = 1 ∞ μ n χ [ x n , b ] ( x ) {\displaystyle f(x)=\sum _{n=1}^{\infty }\mu _{n}\chi _{[x_{n},b]}(x)} <p>where χ<i>A</i> denotes the <a href="/facts/Indicator_function/QEuh04NM">characteristic function</a> of a compact interval <i>A</i>. Then <i>f</i> is a non-decreasing function on [<i>a</i>,<i>b</i>], which is continuous except for jump discontinuities at <i>x</i><i>n</i> for <i>n</i> ≥ 1. In the case of finitely many jump discontinuities, <i>f</i> is a <a href="/facts/Step_function/VnrEjo4B">step function</a>. The examples above are generalised step functions; they are very special cases of what are called jump functions or saltus-functions.<a class="footnote-ref" id="fnref:10" href="#fn:10"><sup>10</sup></a><a class="footnote-ref" id="fnref:11" href="#fn:11"><sup>11</sup></a> </p><p>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 [<i>a</i>,<i>b</i>] can be finite or have ∞ or −∞ as endpoints. </p><p>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 <i>x</i><i>n</i> (<i>n</i> ≥ 1) lie in (<i>a</i>, <i>b</i>) and take λ1, λ2, λ3, ... and μ1, μ2, μ3, ... non-negative with finite sum and with λ<i>n</i> + μ<i>n</i> > 0 for each <i>n</i>. Define </p> f n ( x ) = 0 {\displaystyle f_{n}(x)=0\,\,} for x < x n , f n ( x n ) = λ n , f n ( x ) = λ n + μ n {\displaystyle \,\,x<x_{n},\,\,f_{n}(x_{n})=\lambda _{n},\,\,f_{n}(x)=\lambda _{n}+\mu _{n}\,\,} for x > x n . {\displaystyle \,\,x>x_{n}.} <p>Then the jump function, or saltus-function, defined by </p> f ( x ) = ∑ n = 1 ∞ f n ( x ) = ∑ x n ≤ x λ n + ∑ x n < x μ n , {\displaystyle f(x)=\,\,\sum _{n=1}^{\infty }f_{n}(x)=\,\,\sum _{x_{n}\leq x}\lambda _{n}+\sum _{x_{n}<x}\mu _{n},} <p>is non-decreasing on [<i>a</i>, <i>b</i>] and is continuous except for <a href="/facts/Jump_discontinuity/TGLMsQom">jump discontinuities</a> at <i>x</i><i>n</i> for <i>n</i> ≥ 1.<a class="footnote-ref" id="fnref:12" href="#fn:12"><sup>12</sup></a><a class="footnote-ref" id="fnref:13" href="#fn:13"><sup>13</sup></a><a class="footnote-ref" id="fnref:14" href="#fn:14"><sup>14</sup></a><a class="footnote-ref" id="fnref:15" href="#fn:15"><sup>15</sup></a> </p><p>To prove this, note that sup |<i>f</i><i>n</i>| = λ<i>n</i> + μ<i>n</i>, so that Σ <i>f</i><i>n</i> converges uniformly to <i>f</i>. Passing to the limit, it follows that </p> f ( x n ) − f ( x n − 0 ) = λ n , f ( x n + 0 ) − f ( x n ) = μ n , {\displaystyle f(x_{n})-f(x_{n}-0)=\lambda _{n},\,\,\,f(x_{n}+0)-f(x_{n})=\mu _{n},\,\,\,} and f ( x ± 0 ) = f ( x ) {\displaystyle \,\,f(x\pm 0)=f(x)} <p>if <i>x</i> is not one of the <i>x</i><i>n</i>'s.<a class="footnote-ref" id="fnref:16" href="#fn:16"><sup>16</sup></a> </p><p>Conversely, by a differentiation theorem of <a href="/facts/Henri_Lebesgue/ysngVGXc">Lebesgue</a>, the jump function <i>f</i> is uniquely determined by the properties:<a class="footnote-ref" id="fnref:17" href="#fn:17"><sup>17</sup></a> (1) being non-decreasing and non-positive; (2) having given jump data at its points of discontinuity <i>x</i><i>n</i>; (3) satisfying the boundary condition <i>f</i>(<i>a</i>) = 0; and (4) having zero derivative <a href="/facts/Null_set/kMudL8Mn">almost everywhere</a>. </p> <table><tbody><tr><th>Proof that a jump function has zero derivative almost everywhere.</th></tr><tr><td><p>Property (4) can be checked following Riesz & Sz.-Nagy (1990), Rubel (1963) and Komornik (2016). Without loss of generality, it can be assumed that <i>f</i> is a non-negative jump function defined on the compact [<i>a</i>,<i>b</i>], with discontinuities only in (<i>a</i>,<i>b</i>).</p><p>Note that an open set <i>U</i> of (<i>a</i>,<i>b</i>) is canonically the disjoint union of at most countably many open intervals <i>I</i><i>m</i>; that allows the total length to be computed ℓ(<i>U</i>)= Σ ℓ(<i>I</i><i>m</i>). Recall that a null set <i>A</i> is a subset such that, for any arbitrarily small ε' > 0, there is an open <i>U</i> containing <i>A</i> with ℓ(<i>U</i>) < ε'. A crucial property of length is that, if <i>U</i> and <i>V</i> are open in (<i>a</i>,<i>b</i>), then ℓ(<i>U</i>) + ℓ(<i>V</i>) = ℓ(<i>U</i> ∪ <i>V</i>) + ℓ(<i>U</i> ∩ <i>V</i>).<a class="footnote-ref" id="fnref:18" href="#fn:18"><sup>18</sup></a> It implies immediately that the union of two null sets is null; and that a finite or countable set is null.<a class="footnote-ref" id="fnref:19" href="#fn:19"><sup>19</sup></a><a class="footnote-ref" id="fnref:20" href="#fn:20"><sup>20</sup></a></p><p>Proposition 1. For <i>c</i> > 0 and a normalised non-negative jump function <i>f</i>, let <i>U</i>c(<i>f</i>) be the set of points <i>x</i> such that</p> f ( t ) − f ( s ) t − s > c {\displaystyle {f(t)-f(s) \over t-s}>c} <p>for some <i>s</i>, <i>t</i> with <i>s</i> < <i>x</i> < <i>t</i>. Then<i>U</i><i>c</i>(<i>f</i>) is open and has total length ℓ(<i>U</i><i>c</i>(<i>f</i>)) ≤ 4 <i>c</i>−1 (<i>f</i>(<i>b</i>) – <i>f</i>(<i>a</i>)).</p><p>Note that <i>U</i>c(<i>f</i>) consists the points <i>x</i> where the slope of <i>h</i> is greater that <i>c</i> near <i>x</i>. By definition <i>U</i>c(<i>f</i>) is an open subset of (<i>a</i>, <i>b</i>), so can be written as a disjoint union of at most countably many open intervals <i>I</i><i>k</i> = (<i>a</i><i>k</i>, <i>b</i><i>k</i>). Let <i>J</i><i>k</i> be an interval with closure in <i>I</i><i>k</i> and ℓ(<i>J</i><i>k</i>) = ℓ(<i>I</i><i>k</i>)/2. By compactness, there are finitely many open intervals of the form (<i>s</i>,<i>t</i>) covering the closure of <i>J</i><i>k</i>. 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 (<i>s</i><i>k</i>,1,<i>t</i><i>k</i>,1), (<i>s</i><i>k</i>,2,<i>t</i><i>k</i>,2), ... only intersecting at consecutive intervals.<a class="footnote-ref" id="fnref:21" href="#fn:21"><sup>21</sup></a> Hence</p> ℓ ( J k ) ≤ ∑ m ( t k , m − s k , m ) ≤ ∑ m c − 1 ( f ( t k , m ) − f ( s k , m ) ) ≤ 2 c − 1 ( f ( b k ) − f ( a k ) ) . {\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})).} <p>Finally sum both sides over <i>k</i>.<a class="footnote-ref" id="fnref:22" href="#fn:22"><sup>22</sup></a><a class="footnote-ref" id="fnref:23" href="#fn:23"><sup>23</sup></a></p><p>Proposition 2. If <i>f</i> is a jump function, then <i>f</i> '(<i>x</i>) = 0 almost everywhere.</p><p>To prove this, define</p> D f ( x ) = lim sup s , t → x , s < x < t f ( t ) − f ( s ) t − s , {\displaystyle Df(x)=\limsup _{s,t\rightarrow x,\,\,s<x<t}{f(t)-f(s) \over t-s},} <p>a variant of the <a href="/facts/Dini_derivative/WqDbXenx">Dini derivative</a> of <i>f</i>. It will suffice to prove that for any fixed <i>c</i> > 0, the Dini derivative satisfies <i>D</i><i>f</i>(<i>x</i>) ≤ <i>c</i> <a href="/facts/Almost_everywhere/1quRFtJE">almost everywhere</a>, i.e. on a <a href="/facts/Null_set/kMudL8Mn">null set</a>.</p><p>Choose ε > 0, arbitrarily small. Starting from the definition of the jump function <i>f</i> = Σ <i>f</i><i>n</i>, write <i>f</i> = <i>g</i> + <i>h</i> with <i>g</i> = Σ<i>n</i>≤<i>N</i> <i>f</i><i>n</i> and <i>h</i> = Σ<i>n</i>><i>N</i> <i>f</i><i>n</i> where <i>N</i> ≥ 1. Thus <i>g</i> is a step function having only finitely many discontinuities at <i>x</i><i>n</i> for <i>n</i> ≤ <i>N</i> and <i>h</i> is a non-negative jump function. It follows that <i>D</i><i>f</i> = <i>g</i>' +<i>D</i><i>h</i> = <i>D</i><i>h</i> except at the <i>N</i> points of discontinuity of <i>g</i>. Choosing <i>N</i> sufficiently large so that Σ<i>n</i>><i>N</i> λ<i>n</i> + μ<i>n</i> < ε, it follows that <i>h</i> is a jump function such that <i>h</i>(<i>b</i>) − <i>h</i>(<i>a</i>) < ε and <i>Dh</i> ≤ <i>c</i> off an open set with length less than 4ε/<i>c</i>.</p><p>By construction <i>Df</i> ≤ <i>c</i> off an open set with length less than 4ε/<i>c</i>. Now set ε' = 4ε/<i>c</i> — then ε' and <i>c</i> are arbitrarily small and <i>Df</i> ≤ <i>c</i> off an open set of length less than ε'. Thus <i>Df</i> ≤ <i>c</i> almost everywhere. Since <i>c</i> could be taken arbitrarily small, <i>Df</i> and hence also <i>f</i> ' must vanish almost everywhere.<a class="footnote-ref" id="fnref:24" href="#fn:24"><sup>24</sup></a><a class="footnote-ref" id="fnref:25" href="#fn:25"><sup>25</sup></a></p></td></tr></tbody></table> <p>As explained in Riesz & Sz.-Nagy (1990), every non-decreasing non-negative function <i>F</i> can be decomposed uniquely as a sum of a jump function <i>f</i> and a continuous monotone function <i>g</i>: the jump function <i>f</i> is constructed by using the jump data of the original monotone function <i>F</i> and it is easy to check that <i>g</i> = <i>F</i> − <i>f</i> is continuous and monotone.<a class="footnote-ref" id="fnref:26" href="#fn:26"><sup>26</sup></a> </p> <h2 id="see-also">See also</h2> <ul><li><a href="/facts/Continuous_function/sKbl02pB">Continuous function</a> – Mathematical function with no sudden changes</li> <li><a href="/facts/Bounded_variation/tIjQTpjh">Bounded variation</a> – Real function with finite total variation</li> <li><a href="/facts/Monotone_function/MjQK0YL2">Monotone function</a></li></ul> <h2 id="notes">Notes</h2> <h2 id="bibliography">Bibliography</h2> <ul><li><a href="/facts/Tom_Apostol/S6vGtJWN">Apostol, Tom M.</a> (1957). <a href="https://archive.org/details/in.ernet.dli.2015.141035/page/n173/mode/2up"><i>Mathematical Analysis: a Modern Approach to Advanced Calculus</i></a>. <a href="/facts/Addison-Wesley/A7UFATEG">Addison-Wesley</a>. pp. 162–163. <a href="/facts/MR_(identifier)/uP137L11">MR</a> <a href="https://mathscinet.ams.org/mathscinet-getitem?mr=0087718">0087718</a>.</li> <li><a href="/facts/Ralph_P._Boas_Jr./41W8ymlw">Boas, Ralph P. Jr.</a> (1961). <a href="http://matwbn.icm.edu.pl/ksiazki/cm/cm8/cm8115.pdf">"Differentiability of jump functions"</a> (PDF). <i>Colloq. Math</i>. 8: 81–82. <a href="/facts/Doi_(identifier)/muM9Etpq">doi</a>:<a href="https://doi.org/10.4064%2Fcm-8-1-81-82">10.4064/cm-8-1-81-82</a>. <a href="/facts/MR_(identifier)/uP137L11">MR</a> <a href="https://mathscinet.ams.org/mathscinet-getitem?mr=0126513">0126513</a>.</li> <li>Boas, Ralph P. Jr. (1996). "22. Monotonic functions". <a href="https://www.cambridge.org/core/books/primer-of-real-functions/097C383F0BF28D65F3D32D9A9924548B"><i>A Primer of Real Functions</i></a>. Carus Mathematical Monographs. Vol. 13 (Fourth ed.). <a href="/facts/Mathematical_Association_of_America/G3BkG5YX">MAA</a>. pp. 158–174. <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 978-1-61444-013-0. (subscription required)</li> <li><a href="/facts/John_Charles_Burkill/sv7N0Ury">Burkill, J. C.</a> (1951). <a href="https://archive.org/details/lebesgueintegral0000burk"><i>The Lebesgue integral</i></a>. Cambridge Tracts in Mathematics and Mathematical Physics. Vol. 40. <a href="/facts/Cambridge_University_Press/fgEBSSRq">Cambridge University Press</a>. <a href="/facts/MR_(identifier)/uP137L11">MR</a> <a href="https://mathscinet.ams.org/mathscinet-getitem?mr=0045196">0045196</a>.</li> <li>Edgar, Gerald A. (2008). "Topological Dimension". <i>Measure, topology, and fractal geometry</i>. Undergraduate Texts in Mathematics (Second ed.). <a href="/facts/Springer-Verlag/nAesf6nT">Springer-Verlag</a>. pp. 85–114. <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 978-0-387-74748-4. <a href="/facts/MR_(identifier)/uP137L11">MR</a> <a href="https://mathscinet.ams.org/mathscinet-getitem?mr=2356043">2356043</a>.</li> <li><a href="/facts/Bernard_Russell_Gelbaum/yerxFrmh">Gelbaum, Bernard R.</a>; Olmsted, John M. H. (1964), <a href="https://books.google.com/books?id=D_XBAgAAQBAJ&pg=PA28">"18: A monotonic function whose points of discontinuity form an arbitrary countable (possibly dense) set"</a>, <i>Counterexamples in Analysis</i>, The Mathesis Series, San Francisco, London, Amsterdam: Holden-Day, p. 28, <a href="/facts/MR_(identifier)/uP137L11">MR</a> <a href="https://mathscinet.ams.org/mathscinet-getitem?mr=0169961">0169961</a>; reprinted by Dover, 2003</li> <li><a href="/facts/E._W._Hobson/fRSpQ5nG">Hobson, Ernest W.</a> (1907). <a href="https://archive.org/details/theoryfunctions00hobsgoog/page/244/mode/2up"><i>The Theory of Functions of a Real Variable and their Fourier's Series</i></a>. <a href="/facts/Cambridge_University_Press/fgEBSSRq">Cambridge University Press</a>. p. 245.</li> <li>Komornik, Vilmos (2016). "4. Monotone Functions". <i>Lectures on functional analysis and the Lebesgue integral</i>. Universitext. <a href="/facts/Springer-Verlag/nAesf6nT">Springer-Verlag</a>. pp. 151–164. <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 978-1-4471-6810-2. <a href="/facts/MR_(identifier)/uP137L11">MR</a> <a href="https://mathscinet.ams.org/mathscinet-getitem?mr=3496354">3496354</a>.</li> <li>Lipiński, J. S. (1961). <a href="http://matwbn.icm.edu.pl/ksiazki/cm/cm8/cm8136.pdf">"Une simple démonstration du théorème sur la dérivée d'une fonction de sauts"</a> (PDF). <i>Colloq. Math.</i> (in French). 8 (2): 251–255. <a href="/facts/Doi_(identifier)/muM9Etpq">doi</a>:<a href="https://doi.org/10.4064%2Fcm-8-2-251-255">10.4064/cm-8-2-251-255</a>. <a href="/facts/MR_(identifier)/uP137L11">MR</a> <a href="https://mathscinet.ams.org/mathscinet-getitem?mr=0158036">0158036</a>.</li> <li><a href="/facts/Stanis%25C5%2582aw_%25C5%2581ojasiewicz/0HS9VAYn">Łojasiewicz, Stanisław</a> (1988). "1. Functions of bounded variation". <a href="https://archive.org/details/introductiontoth0000ojas"><i>An introduction to the theory of real functions</i></a>. Translated by G. H. Lawden (Third ed.). Chichester: <a href="/facts/John_Wiley_%2526_Sons/53g3LqJc">John Wiley & Sons</a>. pp. 10–30. <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 0-471-91414-2. <a href="/facts/MR_(identifier)/uP137L11">MR</a> <a href="https://mathscinet.ams.org/mathscinet-getitem?mr=0952856">0952856</a>.</li> <li><a href="/facts/Isidor_Natanson/PsKYIsnK">Natanson, Isidor P.</a> (1955), "III. Functions of finite variation. The Stieltjes integral", <a href="https://archive.org/details/theoryoffunction00nat"><i>Theory of functions of a real variable</i></a>, vol. 1, translated by Leo F. Boron, New York: Frederick Ungar, pp. 204–206, <a href="/facts/MR_(identifier)/uP137L11">MR</a> <a href="https://mathscinet.ams.org/mathscinet-getitem?mr=0067952">0067952</a></li> <li><a href="/facts/Miron_Nicolescu/aVNbFAaP">Nicolescu, M.</a>; Dinculeanu, N.; <a href="/facts/Solomon_Marcus/tktu0geJ">Marcus, S.</a> (1971), <i>Analizǎ Matematică</i> (in Romanian), vol. I (4th ed.), Bucharest: Editura Didactică şi Pedagogică, p. 783, <a href="/facts/MR_(identifier)/uP137L11">MR</a> <a href="https://mathscinet.ams.org/mathscinet-getitem?mr=0352352">0352352</a></li> <li>Olmsted, John M. H. (1959), <i>Real Variables: An Introduction to the Theory of Functions</i>, The Appleton-Century Mathematics Series, New York: Appleton-Century-Crofts, Exercise 29, p. 59, <a href="/facts/MR_(identifier)/uP137L11">MR</a> <a href="https://mathscinet.ams.org/mathscinet-getitem?mr=0117304">0117304</a></li> <li><a href="/facts/Frigyes_Riesz/EN47QveQ">Riesz, Frigyes</a>; <a href="/facts/Bela_Szokefalvi-Nagy/uUlS3KJf">Sz.-Nagy, Béla</a> (1990). "Saltus Functions". <i>Functional analysis</i>. Translated by Leo F. Boron. Dover Books. pp. 13–15. <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 0-486-66289-6. <a href="/facts/MR_(identifier)/uP137L11">MR</a> <a href="https://mathscinet.ams.org/mathscinet-getitem?mr=1068530">1068530</a>. Reprint of the 1955 original.</li> <li><a href="/facts/Stanis%25C5%2582aw_Saks/Duvx15MM">Saks, Stanisław</a> (1937). <a href="http://matwbn.icm.edu.pl/ksiazki/mon/mon07/mon0703.pdf">"III. Functions of bounded variation and the Lebesgue-Stieltjes integral"</a> (PDF). <i>Theory of the integral</i>. Monografie Matematyczne. Vol. VII. Translated by L. C. Young. New York: G. E. Stechert. pp. 96–98.</li> <li><a href="/facts/Lee_Albert_Rubel/2puIcqNW">Rubel, Lee A.</a> (1963). <a href="http://matwbn.icm.edu.pl/ksiazki/cm/cm10/cm10138.pdf">"Differentiability of monotonic functions"</a> (PDF). <i>Colloq. Math</i>. 10 (2): 277–279. <a href="/facts/Doi_(identifier)/muM9Etpq">doi</a>:<a href="https://doi.org/10.4064%2Fcm-10-2-277-279">10.4064/cm-10-2-277-279</a>. <a href="/facts/MR_(identifier)/uP137L11">MR</a> <a href="https://mathscinet.ams.org/mathscinet-getitem?mr=0154954">0154954</a>.</li> <li><a href="/facts/Walter_Rudin/9mxRAECY">Rudin, Walter</a> (1964), <i>Principles of Mathematical Analysis</i> (2nd ed.), New York: McGraw-Hill, <a href="/facts/MR_(identifier)/uP137L11">MR</a> <a href="https://mathscinet.ams.org/mathscinet-getitem?mr=0166310">0166310</a></li> <li><a href="/facts/John_von_Neumann/aUk6urof">von Neumann, John</a> (1950). "IX. Monotonic Functions". <i>Functional Operators. I. Measures and Integrals</i>. Annals of Mathematics Studies. Vol. 21. <a href="/facts/Princeton_University_Press/kwvFlKEj">Princeton University Press</a>. pp. 63–82. <a href="/facts/Doi_(identifier)/muM9Etpq">doi</a>:<a href="https://doi.org/10.1515%2F9781400881895">10.1515/9781400881895</a>. <a href="/facts/ISBN_(identifier)/15AdSPa9">ISBN</a> 978-1-4008-8189-5. <a href="/facts/MR_(identifier)/uP137L11">MR</a> <a href="https://mathscinet.ams.org/mathscinet-getitem?mr=0032011">0032011</a>. {{cite book}}: ISBN / Date incompatibility (help)</li> <li>Young, William Henry; Young, Grace Chisholm (1911). <a href="https://londmathsoc.onlinelibrary.wiley.com/doi/epdf/10.1112/plms/s2-9.1.325">"On the Existence of a Differential Coefficient"</a>. <i><a href="/facts/Proc._London_Math._Soc./jdcf3vhX">Proc. London Math. Soc.</a></i> 2. 9 (1): 325–335. <a href="/facts/Doi_(identifier)/muM9Etpq">doi</a>:<a href="https://doi.org/10.1112%2Fplms%2Fs2-9.1.325">10.1112/plms/s2-9.1.325</a>.</li></ul> <h2 id="references">References</h2> <ol> <li id="fn:1"><p>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. <a href="http://www.numdam.org/item/THESE_1929__102__1_0.pdf" target="_blank">http://www.numdam.org/item/THESE_1929__102__1_0.pdf</a> <a href="#fnref:1" class="footnote-back-ref">↩</a></p></li> <li id="fn:2"><p>Jean Gaston Darboux, Mémoire sur les fonctions discontinues, Annales Scientifiques de l'École Normale Supérieure, 2-ème série, t. IV, 1875, Chap VI. <a href="/wiki/Jean_Gaston_Darboux" target="_blank">/wiki/Jean_Gaston_Darboux</a> <a href="#fnref:2" class="footnote-back-ref">↩</a></p></li> <li id="fn:3"><p>Nicolescu, Dinculeanu & Marcus 1971, p. 213. - Nicolescu, M.; Dinculeanu, N.; Marcus, S. (1971), Analizǎ Matematică (in Romanian), vol. I (4th ed.), Bucharest: Editura Didactică şi Pedagogică, p. 783, MR 0352352 <a href="https://mathscinet.ams.org/mathscinet-getitem?mr=0352352" target="_blank">https://mathscinet.ams.org/mathscinet-getitem?mr=0352352</a> <a href="#fnref:3" class="footnote-back-ref">↩</a></p></li> <li id="fn:4"><p>Rudin 1964, Def. 4.26, pp. 81–82. - Rudin, Walter (1964), Principles of Mathematical Analysis (2nd ed.), New York: McGraw-Hill, MR 0166310 <a href="https://mathscinet.ams.org/mathscinet-getitem?mr=0166310" target="_blank">https://mathscinet.ams.org/mathscinet-getitem?mr=0166310</a> <a href="#fnref:4" class="footnote-back-ref">↩</a></p></li> <li id="fn:5"><p>Rudin 1964, Corollary, p. 83. - Rudin, Walter (1964), Principles of Mathematical Analysis (2nd ed.), New York: McGraw-Hill, MR 0166310 <a href="https://mathscinet.ams.org/mathscinet-getitem?mr=0166310" target="_blank">https://mathscinet.ams.org/mathscinet-getitem?mr=0166310</a> <a href="#fnref:5" class="footnote-back-ref">↩</a></p></li> <li id="fn:6"><p>Nicolescu, Dinculeanu & Marcus 1971, p. 213. - Nicolescu, M.; Dinculeanu, N.; Marcus, S. (1971), Analizǎ Matematică (in Romanian), vol. I (4th ed.), Bucharest: Editura Didactică şi Pedagogică, p. 783, MR 0352352 <a href="https://mathscinet.ams.org/mathscinet-getitem?mr=0352352" target="_blank">https://mathscinet.ams.org/mathscinet-getitem?mr=0352352</a> <a href="#fnref:6" class="footnote-back-ref">↩</a></p></li> <li id="fn:7"><p>Apostol 1957, pp. 162–3. - Apostol, Tom M. (1957). Mathematical Analysis: a Modern Approach to Advanced Calculus. Addison-Wesley. pp. 162–163. MR 0087718. <a href="https://archive.org/details/in.ernet.dli.2015.141035/page/n173/mode/2up" target="_blank">https://archive.org/details/in.ernet.dli.2015.141035/page/n173/mode/2up</a> <a href="#fnref:7" class="footnote-back-ref">↩</a></p></li> <li id="fn:8"><p>Hobson 1907, p. 245. - Hobson, Ernest W. (1907). The Theory of Functions of a Real Variable and their Fourier's Series. Cambridge University Press. p. 245. <a href="https://archive.org/details/theoryfunctions00hobsgoog/page/244/mode/2up" target="_blank">https://archive.org/details/theoryfunctions00hobsgoog/page/244/mode/2up</a> <a href="#fnref:8" class="footnote-back-ref">↩</a></p></li> <li id="fn:9"><p>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 [ a n , b n ] ⊆ [ a n + 1 , b n + 1 ] {\displaystyle \left[a_{n},b_{n}\right]\subseteq \left[a_{n+1},b_{n+1}\right]} for all n {\displaystyle n} <a href="/wiki/Disjoint_sets" target="_blank">/wiki/Disjoint_sets</a> <a href="#fnref:9" class="footnote-back-ref">↩</a></p></li> <li id="fn:10"><p>Apostol 1957. - Apostol, Tom M. (1957). Mathematical Analysis: a Modern Approach to Advanced Calculus. Addison-Wesley. pp. 162–163. MR 0087718. <a href="https://archive.org/details/in.ernet.dli.2015.141035/page/n173/mode/2up" target="_blank">https://archive.org/details/in.ernet.dli.2015.141035/page/n173/mode/2up</a> <a href="#fnref:10" class="footnote-back-ref">↩</a></p></li> <li id="fn:11"><p>Riesz & Sz.-Nagy 1990. - Riesz, Frigyes; Sz.-Nagy, Béla (1990). "Saltus Functions". Functional analysis. Translated by Leo F. Boron. Dover Books. pp. 13–15. ISBN 0-486-66289-6. MR 1068530. <a href="https://mathscinet.ams.org/mathscinet-getitem?mr=1068530" target="_blank">https://mathscinet.ams.org/mathscinet-getitem?mr=1068530</a> <a href="#fnref:11" class="footnote-back-ref">↩</a></p></li> <li id="fn:12"><p>Riesz & Sz.-Nagy 1990, pp. 13–15 - Riesz, Frigyes; Sz.-Nagy, Béla (1990). "Saltus Functions". Functional analysis. Translated by Leo F. Boron. Dover Books. pp. 13–15. ISBN 0-486-66289-6. MR 1068530. <a href="https://mathscinet.ams.org/mathscinet-getitem?mr=1068530" target="_blank">https://mathscinet.ams.org/mathscinet-getitem?mr=1068530</a> <a href="#fnref:12" class="footnote-back-ref">↩</a></p></li> <li id="fn:13"><p>Saks 1937. - Saks, Stanisław (1937). "III. Functions of bounded variation and the Lebesgue-Stieltjes integral" (PDF). Theory of the integral. Monografie Matematyczne. Vol. VII. Translated by L. C. Young. New York: G. E. Stechert. pp. 96–98. <a href="http://matwbn.icm.edu.pl/ksiazki/mon/mon07/mon0703.pdf" target="_blank">http://matwbn.icm.edu.pl/ksiazki/mon/mon07/mon0703.pdf</a> <a href="#fnref:13" class="footnote-back-ref">↩</a></p></li> <li id="fn:14"><p>Natanson 1955. - Natanson, Isidor P. (1955), "III. Functions of finite variation. The Stieltjes integral", Theory of functions of a real variable, vol. 1, translated by Leo F. Boron, New York: Frederick Ungar, pp. 204–206, MR 0067952 <a href="https://archive.org/details/theoryoffunction00nat" target="_blank">https://archive.org/details/theoryoffunction00nat</a> <a href="#fnref:14" class="footnote-back-ref">↩</a></p></li> <li id="fn:15"><p>Łojasiewicz 1988. - Łojasiewicz, Stanisław (1988). "1. Functions of bounded variation". An introduction to the theory of real functions. Translated by G. H. Lawden (Third ed.). Chichester: John Wiley & Sons. pp. 10–30. ISBN 0-471-91414-2. MR 0952856. <a href="https://archive.org/details/introductiontoth0000ojas" target="_blank">https://archive.org/details/introductiontoth0000ojas</a> <a href="#fnref:15" class="footnote-back-ref">↩</a></p></li> <li id="fn:16"><p>Riesz & Sz.-Nagy 1990, pp. 13–15 - Riesz, Frigyes; Sz.-Nagy, Béla (1990). "Saltus Functions". Functional analysis. Translated by Leo F. Boron. Dover Books. pp. 13–15. ISBN 0-486-66289-6. MR 1068530. <a href="https://mathscinet.ams.org/mathscinet-getitem?mr=1068530" target="_blank">https://mathscinet.ams.org/mathscinet-getitem?mr=1068530</a> <a href="#fnref:16" class="footnote-back-ref">↩</a></p></li> <li id="fn:17"><p>For more details, see Riesz & Sz.-Nagy 1990 Young & Young 1911 von Neumann 1950 Boas 1961 Lipiński 1961 Rubel 1963 Komornik 2016 - Riesz, Frigyes; Sz.-Nagy, Béla (1990). "Saltus Functions". Functional analysis. Translated by Leo F. Boron. Dover Books. pp. 13–15. ISBN 0-486-66289-6. MR 1068530. <a href="https://mathscinet.ams.org/mathscinet-getitem?mr=1068530" target="_blank">https://mathscinet.ams.org/mathscinet-getitem?mr=1068530</a> <a href="#fnref:17" class="footnote-back-ref">↩</a></p></li> <li id="fn:18"><p>Burkill 1951, pp. 10−11. - Burkill, J. C. (1951). The Lebesgue integral. Cambridge Tracts in Mathematics and Mathematical Physics. Vol. 40. Cambridge University Press. MR 0045196. <a href="https://archive.org/details/lebesgueintegral0000burk" target="_blank">https://archive.org/details/lebesgueintegral0000burk</a> <a href="#fnref:18" class="footnote-back-ref">↩</a></p></li> <li id="fn:19"><p>Rubel 1963 - Rubel, Lee A. (1963). "Differentiability of monotonic functions" (PDF). Colloq. Math. 10 (2): 277–279. doi:10.4064/cm-10-2-277-279. MR 0154954. <a href="http://matwbn.icm.edu.pl/ksiazki/cm/cm10/cm10138.pdf" target="_blank">http://matwbn.icm.edu.pl/ksiazki/cm/cm10/cm10138.pdf</a> <a href="#fnref:19" class="footnote-back-ref">↩</a></p></li> <li id="fn:20"><p>Komornik 2016 - Komornik, Vilmos (2016). "4. Monotone Functions". Lectures on functional analysis and the Lebesgue integral. Universitext. Springer-Verlag. pp. 151–164. ISBN 978-1-4471-6810-2. MR 3496354. <a href="https://mathscinet.ams.org/mathscinet-getitem?mr=3496354" target="_blank">https://mathscinet.ams.org/mathscinet-getitem?mr=3496354</a> <a href="#fnref:20" class="footnote-back-ref">↩</a></p></li> <li id="fn:21"><p>This is a simple example of how Lebesgue covering dimension applies in one real dimension; see for example Edgar (2008). <a href="/wiki/Lebesgue_covering_dimension" target="_blank">/wiki/Lebesgue_covering_dimension</a> <a href="#fnref:21" class="footnote-back-ref">↩</a></p></li> <li id="fn:22"><p>Rubel 1963 - Rubel, Lee A. (1963). "Differentiability of monotonic functions" (PDF). Colloq. Math. 10 (2): 277–279. doi:10.4064/cm-10-2-277-279. MR 0154954. <a href="http://matwbn.icm.edu.pl/ksiazki/cm/cm10/cm10138.pdf" target="_blank">http://matwbn.icm.edu.pl/ksiazki/cm/cm10/cm10138.pdf</a> <a href="#fnref:22" class="footnote-back-ref">↩</a></p></li> <li id="fn:23"><p>Komornik 2016 - Komornik, Vilmos (2016). "4. Monotone Functions". Lectures on functional analysis and the Lebesgue integral. Universitext. Springer-Verlag. pp. 151–164. ISBN 978-1-4471-6810-2. MR 3496354. <a href="https://mathscinet.ams.org/mathscinet-getitem?mr=3496354" target="_blank">https://mathscinet.ams.org/mathscinet-getitem?mr=3496354</a> <a href="#fnref:23" class="footnote-back-ref">↩</a></p></li> <li id="fn:24"><p>Rubel 1963 - Rubel, Lee A. (1963). "Differentiability of monotonic functions" (PDF). Colloq. Math. 10 (2): 277–279. doi:10.4064/cm-10-2-277-279. MR 0154954. <a href="http://matwbn.icm.edu.pl/ksiazki/cm/cm10/cm10138.pdf" target="_blank">http://matwbn.icm.edu.pl/ksiazki/cm/cm10/cm10138.pdf</a> <a href="#fnref:24" class="footnote-back-ref">↩</a></p></li> <li id="fn:25"><p>Komornik 2016 - Komornik, Vilmos (2016). "4. Monotone Functions". Lectures on functional analysis and the Lebesgue integral. Universitext. Springer-Verlag. pp. 151–164. ISBN 978-1-4471-6810-2. MR 3496354. <a href="https://mathscinet.ams.org/mathscinet-getitem?mr=3496354" target="_blank">https://mathscinet.ams.org/mathscinet-getitem?mr=3496354</a> <a href="#fnref:25" class="footnote-back-ref">↩</a></p></li> <li id="fn:26"><p>Riesz & Sz.-Nagy 1990, pp. 13–15 - Riesz, Frigyes; Sz.-Nagy, Béla (1990). "Saltus Functions". Functional analysis. Translated by Leo F. Boron. Dover Books. pp. 13–15. ISBN 0-486-66289-6. MR 1068530. <a href="https://mathscinet.ams.org/mathscinet-getitem?mr=1068530" target="_blank">https://mathscinet.ams.org/mathscinet-getitem?mr=1068530</a> <a href="#fnref:26" class="footnote-back-ref">↩</a></p></li> </ol>