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 [math]\displaystyle{ f\left(x^-\right) := \lim_{z \nearrow x} f(z) = \lim_{\stackrel{h \to 0}{h \gt 0}} f(x-h) }[/math] and denote the limit from the right by [math]\displaystyle{ f\left(x^+\right) := \lim_{z \searrow x} f(z) = \lim_{\stackrel{h \to 0}{h \gt 0}} f(x+h). }[/math]
If [math]\displaystyle{ f\left(x^+\right) }[/math] and [math]\displaystyle{ f\left(x^-\right) }[/math] exist and are finite then the difference [math]\displaystyle{ f\left(x^+\right) - f\left(x^-\right) }[/math] is called the jump[3] of [math]\displaystyle{ f }[/math] at [math]\displaystyle{ x. }[/math]
Consider a real-valued function [math]\displaystyle{ f }[/math] of real variable [math]\displaystyle{ x }[/math] defined in a neighborhood of a point [math]\displaystyle{ x. }[/math] If [math]\displaystyle{ f }[/math] is discontinuous at the point [math]\displaystyle{ x }[/math] 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 [math]\displaystyle{ x }[/math] then the jump at [math]\displaystyle{ x }[/math] is zero. Moreover, if [math]\displaystyle{ f }[/math] is not continuous at [math]\displaystyle{ x, }[/math] the jump can be zero at [math]\displaystyle{ x }[/math] if [math]\displaystyle{ f\left(x^+\right) = f\left(x^-\right) \neq f(x). }[/math]
Precise statement
Let [math]\displaystyle{ f }[/math] be a real-valued monotone function defined on an interval [math]\displaystyle{ I. }[/math] 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 [math]\displaystyle{ f }[/math] be a monotone function defined on an interval [math]\displaystyle{ I. }[/math] 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 [math]\displaystyle{ [a, b]. }[/math][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 [math]\displaystyle{ I := [a, b] }[/math] be an interval and let [math]\displaystyle{ f : I \to \R }[/math] be a non-decreasing function (such as an increasing function). Then for any [math]\displaystyle{ a \lt x \lt b, }[/math] [math]\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). }[/math] Let [math]\displaystyle{ \alpha \gt 0 }[/math] and let [math]\displaystyle{ x_1 \lt x_2 \lt \cdots \lt x_n }[/math] be [math]\displaystyle{ n }[/math] points inside [math]\displaystyle{ I }[/math] at which the jump of [math]\displaystyle{ f }[/math] is greater or equal to [math]\displaystyle{ \alpha }[/math]: [math]\displaystyle{ f\left(x_i^+\right) - f\left(x_i^-\right) \geq \alpha,\ i=1,2,\ldots,n }[/math]
For any [math]\displaystyle{ i=1,2,\ldots,n, }[/math] [math]\displaystyle{ f\left(x_i^+\right) \leq f\left(x_{i+1}^-\right) }[/math] so that [math]\displaystyle{ f\left(x_{i+1}^-\right) - f\left(x_i^+\right) \geq 0. }[/math] Consequently, [math]\displaystyle{ \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} }[/math] and hence [math]\displaystyle{ n \leq \frac{f(b) - f(a)}{\alpha}. }[/math]
Since [math]\displaystyle{ f(b) - f(a) \lt \infty }[/math] we have that the number of points at which the jump is greater than [math]\displaystyle{ \alpha }[/math] is finite (possibly even zero).
Define the following sets: [math]\displaystyle{ S_1: = \left\{x : x \in I, f\left(x^+\right) - f\left(x^-\right) \geq 1\right\}, }[/math] [math]\displaystyle{ S_n: = \left\{x : x \in I, \frac{1}{n} \leq f\left(x^+\right) - f\left(x^-\right) \lt \frac{1}{n-1}\right\},\ n\geq 2. }[/math]
Each set [math]\displaystyle{ S_n }[/math] is finite or the empty set. The union [math]\displaystyle{ S = \bigcup_{n=1}^\infty S_n }[/math] contains all points at which the jump is positive and hence contains all points of discontinuity. Since every [math]\displaystyle{ S_i,\ i=1,2,\ldots }[/math] is at most countable, their union [math]\displaystyle{ S }[/math] is also at most countable.
If [math]\displaystyle{ f }[/math] 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. [math]\displaystyle{ \blacksquare }[/math]
Proof 2
So let [math]\displaystyle{ f : [a, b] \to \R }[/math] is a monotone function and let [math]\displaystyle{ D }[/math] denote the set of all points [math]\displaystyle{ d \in [a, b] }[/math] in the domain of [math]\displaystyle{ f }[/math] at which [math]\displaystyle{ f }[/math] is discontinuous (which is necessarily a jump discontinuity).
Because [math]\displaystyle{ f }[/math] has a jump discontinuity at [math]\displaystyle{ d \in D, }[/math] [math]\displaystyle{ f\left(d^-\right) \neq f\left(d^+\right) }[/math] so there exists some rational number [math]\displaystyle{ y_d \in \Q }[/math] that lies strictly in between [math]\displaystyle{ f\left(d^-\right) \text{ and } f\left(d^+\right) }[/math] (specifically, if [math]\displaystyle{ f \nearrow }[/math] then pick [math]\displaystyle{ y_d \in \Q }[/math] so that [math]\displaystyle{ f\left(d^-\right) \lt y_d \lt f\left(d^+\right) }[/math] while if [math]\displaystyle{ f \searrow }[/math] then pick [math]\displaystyle{ y_d \in \Q }[/math] so that [math]\displaystyle{ f\left(d^-\right) \gt y_d \gt f\left(d^+\right) }[/math] holds).
It will now be shown that if [math]\displaystyle{ d, e \in D }[/math] are distinct, say with [math]\displaystyle{ d \lt e, }[/math] then [math]\displaystyle{ y_d \neq y_e. }[/math] If [math]\displaystyle{ f \nearrow }[/math] then [math]\displaystyle{ d \lt e }[/math] implies [math]\displaystyle{ f\left(d^+\right) \leq f\left(e^-\right) }[/math] so that [math]\displaystyle{ y_d \lt f\left(d^+\right) \leq f\left(e^-\right) \lt y_e. }[/math] If on the other hand [math]\displaystyle{ f \searrow }[/math] then [math]\displaystyle{ d \lt e }[/math] implies [math]\displaystyle{ f\left(d^+\right) \geq f\left(e^-\right) }[/math] so that [math]\displaystyle{ y_d \gt f\left(d^+\right) \geq f\left(e^-\right) \gt y_e. }[/math] Either way, [math]\displaystyle{ y_d \neq y_e. }[/math]
Thus every [math]\displaystyle{ d \in D }[/math] is associated with a unique rational number (said differently, the map [math]\displaystyle{ D \to \Q }[/math] defined by [math]\displaystyle{ d \mapsto y_d }[/math] is injective). Since [math]\displaystyle{ \Q }[/math] is countable, the same must be true of [math]\displaystyle{ D. }[/math] [math]\displaystyle{ \blacksquare }[/math]
Proof of general case
Suppose that the domain of [math]\displaystyle{ f }[/math] (a monotone real-valued function) is equal to a union of countably many closed and bounded intervals; say its domain is [math]\displaystyle{ \bigcup_{n} \left[a_n, b_n\right] }[/math] (no requirements are placed on these closed and bounded intervals[lower-alpha 1]). It follows from the special case proved above that for every index [math]\displaystyle{ n, }[/math] the restriction [math]\displaystyle{ f\big\vert_{\left[a_n, b_n\right]} : \left[a_n, b_n\right] \to \R }[/math] of [math]\displaystyle{ f }[/math] to the interval [math]\displaystyle{ \left[a_n, b_n\right] }[/math] has at most countably many discontinuities; denote this (countable) set of discontinuities by [math]\displaystyle{ D_n. }[/math] If [math]\displaystyle{ f }[/math] has a discontinuity at a point [math]\displaystyle{ x_0 \in \bigcup_{n} \left[a_n, b_n\right] }[/math] in its domain then either [math]\displaystyle{ x_0 }[/math] is equal to an endpoint of one of these intervals (that is, [math]\displaystyle{ x_0 \in \left\{a_1, b_1, a_2, b_2, \ldots\right\} }[/math]) or else there exists some index [math]\displaystyle{ n }[/math] such that [math]\displaystyle{ a_n \lt x_0 \lt b_n, }[/math] in which case [math]\displaystyle{ x_0 }[/math] must be a point of discontinuity for [math]\displaystyle{ f\big\vert_{\left[a_n, b_n\right]} }[/math] (that is, [math]\displaystyle{ x_0 \in D_n }[/math]). Thus the set [math]\displaystyle{ D }[/math] of all points of at which [math]\displaystyle{ f }[/math] is discontinuous is a subset of [math]\displaystyle{ \left\{a_1, b_1, a_2, b_2, \ldots\right\} \cup \bigcup_{n} D_n, }[/math] which is a countable set (because it is a union of countably many countable sets) so that its subset [math]\displaystyle{ D }[/math] 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 [math]\displaystyle{ f }[/math] is an interval [math]\displaystyle{ I }[/math] 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 [math]\displaystyle{ I_n }[/math] with the property that any two consecutive intervals have an endpoint in common: [math]\displaystyle{ I = \cup_{n=1}^\infty I_n. }[/math] If [math]\displaystyle{ I = (a,b] \text{ with } a \geq -\infty }[/math] then [math]\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 }[/math] where [math]\displaystyle{ \left(\alpha_n\right)_{n=1}^{\infty} }[/math] is a strictly decreasing sequence such that [math]\displaystyle{ \alpha_n \rightarrow a. }[/math] In a similar way if [math]\displaystyle{ I = [a,b), \text{ with } b \leq +\infty }[/math] or if [math]\displaystyle{ I = (a,b) \text{ with } -\infty \leq a \lt b \leq \infty. }[/math] In any interval [math]\displaystyle{ I_n, }[/math] 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. [math]\displaystyle{ \blacksquare }[/math]
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
- [math]\displaystyle{ f(x) = \sum_{n=1}^{\infty} \mu_n \chi_{[x_n,b]} (x) }[/math]
where χA 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 xn 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), 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 xn (n ≥ 1) lie in (a, b) and take λ1, λ2, λ3, ... and μ1, μ2, μ3, ... non-negative with finite sum and with λn + μn > 0 for each n. Define
- [math]\displaystyle{ f_n(x)=0\,\, }[/math] for [math]\displaystyle{ \,\, x \lt x_n,\,\, f_n(x_n) = \lambda_n, \,\, f_n(x) = \lambda_n +\mu_n\,\, }[/math] for [math]\displaystyle{ \,\, x \gt x_n. }[/math]
Then the jump function, or saltus-function, defined by
- [math]\displaystyle{ f(x)=\,\,\sum_{n=1}^\infty f_n(x) =\,\, \sum_{x_n\le x} \lambda_n + \sum_{x_n\lt x} \mu_n, }[/math]
is non-decreasing on [a, b] and is continuous except for jump discontinuities at xn for n ≥ 1.[10][11][12][13]
To prove this, note that sup |fn| = λn + μn, so that Σ fn converges uniformly to f. Passing to the limit, it follows that
- [math]\displaystyle{ f(x_n)-f(x_n-0)=\lambda_n,\,\,\, f(x_n+0)-f(x_n)=\mu_n,\,\,\, }[/math] and [math]\displaystyle{ \,\, f(x\pm 0)=f(x) }[/math]
if x is not one of the xn'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 xn; (3) satisfying the boundary condition f(a) = 0; and (4) having zero derivative almost everywhere.
Expand Proof that a jump function has zero derivative almost everywhere.
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As explained in (Riesz Sz.-Nagy), 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
- ↑ 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 [math]\displaystyle{ \left[a_n, b_n\right] \subseteq \left[a_{n+1}, b_{n+1}\right] }[/math] for all [math]\displaystyle{ n }[/math]
References
- ↑ 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.
- ↑ Jean Gaston Darboux, Mémoire sur les fonctions discontinues, Annales Scientifiques de l'École Normale Supérieure, 2-ème série, t. IV, 1875, Chap VI.
- ↑ Jump up to: 3.0 3.1 Nicolescu, Dinculeanu & Marcus 1971, p. 213.
- ↑ Rudin 1964, Def. 4.26, pp. 81–82.
- ↑ Rudin 1964, Corollary, p. 83.
- ↑ Apostol 1957, pp. 162–3.
- ↑ Hobson 1907, p. 245.
- ↑ Apostol 1957.
- ↑ Riesz & Sz.-Nagy 1990.
- ↑ Jump up to: 10.0 10.1 10.2 Riesz & Sz.-Nagy 1990, pp. 13–15
- ↑ Saks 1937.
- ↑ Natanson 1955.
- ↑ Łojasiewicz 1988.
- ↑ For more details, see
- ↑ Burkill 1951, pp. 10−11.
- ↑ Jump up to: 16.0 16.1 16.2 Rubel 1963
- ↑ Jump up to: 17.0 17.1 17.2 Komornik 2016
- ↑ This is a simple example of how Lebesgue covering dimension applies in one real dimension; see for example (Edgar 2008).
Bibliography
- Apostol, Tom M. (1957). Mathematical Analysis: a Modern Approach to Advanced Calculus. Addison-Wesley. pp. 162–163. https://archive.org/details/in.ernet.dli.2015.141035/page/n173/mode/2up.
- Boas, Ralph P. Jr. (1961). "Differentiability of jump functions". Colloq. Math. 8: 81–82. doi:10.4064/cm-8-1-81-82. http://matwbn.icm.edu.pl/ksiazki/cm/cm8/cm8115.pdf.
- Boas, Ralph P. Jr. (1996). "22. Monotonic functions". A Primer of Real Functions. Carus Mathematical Monographs. 13 (Fourth ed.). MAA. pp. 158–174. ISBN 978-1-61444-013-0. https://www.cambridge.org/core/books/primer-of-real-functions/097C383F0BF28D65F3D32D9A9924548B. (subscription required)
- Burkill, J. C. (1951). The Lebesgue integral. Cambridge Tracts in Mathematics and Mathematical Physics. 40. Cambridge University Press. https://archive.org/details/lebesgueintegral0000burk.
- Edgar, Gerald A. (2008). "Topological Dimension". Measure, topology, and fractal geometry. Undergraduate Texts in Mathematics (Second ed.). Springer-Verlag. pp. 85–114. ISBN 978-0-387-74748-4.
- "18: A monotonic function whose points of discontinuity form an arbitrary countable (possibly dense) set", Counterexamples in Analysis, The Mathesis Series, San Francisco, London, Amsterdam: Holden-Day, 1964, p. 28, https://books.google.com/books?id=D_XBAgAAQBAJ&pg=PA28; reprinted by Dover, 2003
- Hobson, Ernest W. (1907). The Theory of Functions of a Real Variable and their Fourier's Series. Cambridge University Press. p. 245. https://archive.org/details/theoryfunctions00hobsgoog/page/244/mode/2up.
- 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.
- Lipiński, J. S. (1961). "Une simple démonstration du théorème sur la dérivée d'une fonction de sauts" (in fr). Colloq. Math. 8 (2): 251–255. doi:10.4064/cm-8-2-251-255. http://matwbn.icm.edu.pl/ksiazki/cm/cm8/cm8136.pdf.
- Łojasiewicz, Stanisław (1988). "1. Functions of bounded variation". An introduction to the theory of real functions (Third ed.). Chichester: John Wiley & Sons. pp. 10–30. ISBN 0-471-91414-2. https://archive.org/details/introductiontoth0000ojas.
- Natanson, Isidor P. (1955), "III. Functions of finite variation. The Stieltjes integral", Theory of functions of a real variable, 1, New York: Frederick Ungar, pp. 204–206, https://archive.org/details/theoryoffunction00nat
- (in ro) Analizǎ Matematică, I (4th ed.), Bucharest: Editura Didactică şi Pedagogică, 1971, p. 783
- Olmsted, John M. H. (1959), Real Variables: An Introduction to the Theory of Functions, The Appleton-Century Mathematics Series, New York: Appleton-Century-Crofts, Exercise 29, p. 59
- Riesz, Frigyes; Sz.-Nagy, Béla (1990). "Saltus Functions". Functional analysis. Dover Books. pp. 13–15. ISBN 0-486-66289-6. Reprint of the 1955 original.
- Saks, Stanisław (1937). "III. Functions of bounded variation and the Lebesgue-Stieltjes integral". Theory of the integral. Monografie Matematyczne. VII. New York: G. E. Stechert. pp. 96–98. http://matwbn.icm.edu.pl/ksiazki/mon/mon07/mon0703.pdf.
- Rubel, Lee A. (1963). "Differentiability of monotonic functions". Colloq. Math. 10 (2): 277–279. doi:10.4064/cm-10-2-277-279. http://matwbn.icm.edu.pl/ksiazki/cm/cm10/cm10138.pdf.
- Principles of Mathematical Analysis (2nd ed.), New York: McGraw-Hill, 1964
- von Neumann, John (1950). "IX. Monotonic Functions". Functional Operators. I. Measures and Integrals. Annals of Mathematics Studies. 21. Princeton University Press. pp. 63–82. doi:10.1515/9781400881895. ISBN 978-1-4008-8189-5.
- Young, William Henry; Young, Grace Chisholm (1911). "On the Existence of a Differential Coefficient". Proc. London Math. Soc.. 2 9 (1): 325–335. doi:10.1112/plms/s2-9.1.325. https://londmathsoc.onlinelibrary.wiley.com/doi/epdf/10.1112/plms/s2-9.1.325.
Original source: https://en.wikipedia.org/wiki/Discontinuities of monotone functions.
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