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]

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 ${\displaystyle f\left(x^{+}\right)=f\left(x^{-}\right)\neq f(x).}$

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.

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.

Two proofs of this special case are given.

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. ${\displaystyle \blacksquare }$

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 ${\displaystyle D.}$ ${\displaystyle \blacksquare }$

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. ${\displaystyle \blacksquare }$

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.

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]

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

• Apostol, Tom M. (1957). Mathematical Analysis: a Modern Approach to Advanced Calculus. Addison-Wesley. pp. 162–163. MR 0087718.
• Boas, Ralph P. Jr. (1961). "Differentiability of jump functions" (PDF). Colloq. Math. 8: 81–82. doi:10.4064/cm-8-1-81-82. MR 0126513.
• Boas, Ralph P. Jr. (1996). "22. Monotonic functions". A Primer of Real Functions. Carus Mathematical Monographs. Vol. 13 (Fourth ed.). MAA. pp. 158–174. ISBN 978-1-61444-013-0. (subscription required)
• Burkill, J. C. (1951). The Lebesgue integral. Cambridge Tracts in Mathematics and Mathematical Physics. Vol. 40. Cambridge University Press. MR 0045196.
• 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. MR 2356043.
• Gelbaum, Bernard R.; Olmsted, John M. H. (1964), "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, p. 28, MR 0169961; 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.
• 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.
• Lipiński, J. S. (1961). "Une simple démonstration du théorème sur la dérivée d'une fonction de sauts" (PDF). Colloq. Math. (in French). 8 (2): 251–255. doi:10.4064/cm-8-2-251-255. MR 0158036.
• 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.
• 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
• Nicolescu, M.; Dinculeanu, N.; Marcus, S. (1971), Analizǎ Matematică (in Romanian), vol. I (4th ed.), Bucharest: Editura Didactică şi Pedagogică, p. 783, MR 0352352
• 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, MR 0117304
• 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. Reprint of the 1955 original.
• 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.
• 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.
• Rudin, Walter (1964), Principles of Mathematical Analysis (2nd ed.), New York: McGraw-Hill, MR 0166310
• von Neumann, John (1950). "IX. Monotonic Functions". Functional Operators. I. Measures and Integrals. Annals of Mathematics Studies. Vol. 21. Princeton University Press. pp. 63–82. doi:10.1515/9781400881895. ISBN 978-1-4008-8189-5. MR 0032011.
• 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.