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Continuous function In mathematics, a continuous function is a function such that a continuous variation (that is a change without jump) of the argument induces a continuous variation of the value of the function. This means that there are no abrupt changes in value ...
s are of utmost importance in
mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
, functions and applications. However, not all functions are
continuous Continuity or continuous may refer to: Mathematics * Continuity (mathematics), the opposing concept to discreteness; common examples include ** Continuous probability distribution or random variable in probability and statistics ** Continuous ...
. If a function is not continuous at a point in its
domain Domain may refer to: Mathematics *Domain of a function, the set of input values for which the (total) function is defined **Domain of definition of a partial function **Natural domain of a partial function **Domain of holomorphy of a function * Do ...
, one says that it has a discontinuity there. The
set Set, The Set, SET or SETS may refer to: Science, technology, and mathematics Mathematics *Set (mathematics), a collection of elements *Category of sets, the category whose objects and morphisms are sets and total functions, respectively Electro ...
of all points of discontinuity of a function may be a
discrete set ] In mathematics, a point ''x'' is called an isolated point of a subset ''S'' (in a topological space ''X'') if ''x'' is an element of ''S'' and there exists a neighborhood of ''x'' which does not contain any other points of ''S''. This is equival ...
, a
dense set In topology and related areas of mathematics, a subset ''A'' of a topological space ''X'' is said to be dense in ''X'' if every point of ''X'' either belongs to ''A'' or else is arbitrarily "close" to a member of ''A'' — for instance, the r ...
, or even the entire domain of the function. This article describes the classification of discontinuities in the simplest case of functions of a single
real Real may refer to: Currencies * Brazilian real (R$) * Central American Republic real * Mexican real * Portuguese real * Spanish real * Spanish colonial real Music Albums * ''Real'' (L'Arc-en-Ciel album) (2000) * ''Real'' (Bright album) (2010) ...
variable Variable may refer to: * Variable (computer science), a symbolic name associated with a value and whose associated value may be changed * Variable (mathematics), a symbol that represents a quantity in a mathematical expression, as used in many ...
taking real values. The
oscillation Oscillation is the repetitive or periodic variation, typically in time, of some measure about a central value (often a point of equilibrium) or between two or more different states. Familiar examples of oscillation include a swinging pendulum ...
of a function at a point quantifies these discontinuities as follows: * in a removable discontinuity, the distance that the value of the function is off by is the oscillation; * in a jump discontinuity, the size of the jump is the oscillation (assuming that the value ''at'' the point lies between these limits of the two sides); * in an essential discontinuity, oscillation measures the failure of a
limit Limit or Limits may refer to: Arts and media * ''Limit'' (manga), a manga by Keiko Suenobu * ''Limit'' (film), a South Korean film * Limit (music), a way to characterize harmony * "Limit" (song), a 2016 single by Luna Sea * "Limits", a 2019 ...
to exist; the limit is constant. A special case is if the function diverges to
infinity Infinity is that which is boundless, endless, or larger than any natural number. It is often denoted by the infinity symbol . Since the time of the ancient Greeks, the philosophical nature of infinity was the subject of many discussions amo ...
or minus
infinity Infinity is that which is boundless, endless, or larger than any natural number. It is often denoted by the infinity symbol . Since the time of the ancient Greeks, the philosophical nature of infinity was the subject of many discussions amo ...
, in which case the
oscillation Oscillation is the repetitive or periodic variation, typically in time, of some measure about a central value (often a point of equilibrium) or between two or more different states. Familiar examples of oscillation include a swinging pendulum ...
is not defined (in the extended real numbers, this is a removable discontinuity).


Classification

For each of the following, consider a real valued function f of a real variable x, defined in a neighborhood of the point x_0 at which f is discontinuous.


Removable discontinuity

Consider the
piecewise In mathematics, a piecewise-defined function (also called a piecewise function, a hybrid function, or definition by cases) is a function defined by multiple sub-functions, where each sub-function applies to a different interval in the domain. Pi ...
function f(x) = \begin x^2 & \text x < 1 \\ 0 & \text x = 1 \\ 2-x & \text x > 1 \end The point x_0 = 1 is a ''removable discontinuity''. For this kind of discontinuity: The
one-sided limit In calculus, a one-sided limit refers to either one of the two limits of a function f(x) of a real variable x as x approaches a specified point either from the left or from the right. The limit as x decreases in value approaching a (x approaches ...
from the negative direction: L^- = \lim_ f(x) and the one-sided limit from the positive direction: L^+ = \lim_ f(x) at x_0 ''both'' exist, are finite, and are equal to L = L^- = L^+. In other words, since the two one-sided limits exist and are equal, the limit L of f(x) as x approaches x_0 exists and is equal to this same value. If the actual value of f\left(x_0\right) is ''not'' equal to L, then x_0 is called a . This discontinuity can be removed to make f continuous at x_0, or more precisely, the function g(x) = \begin f(x) & x \neq x_0 \\ L & x = x_0 \end is continuous at x = x_0. The term ''removable discontinuity'' is sometimes broadened to include a
removable singularity In complex analysis, a removable singularity of a holomorphic function is a point at which the function is undefined, but it is possible to redefine the function at that point in such a way that the resulting function is regular in a neighbourh ...
, in which the limits in both directions exist and are equal, while the function is
undefined Undefined may refer to: Mathematics * Undefined (mathematics), with several related meanings ** Indeterminate form, in calculus Computing * Undefined behavior, computer code whose behavior is not specified under certain conditions * Undefined ...
at the point x_0. This use is an abuse of terminology because continuity and discontinuity of a function are concepts defined only for points in the function's domain.


Jump discontinuity

Consider the function f(x) = \begin x^2 & \mbox x < 1 \\ 0 & \mbox x = 1 \\ 2 - (x-1)^2 & \mbox x > 1 \end Then, the point x_0 = 1 is a '. In this case, a single limit does not exist because the one-sided limits, L^- and L^+ exist and are finite, but are not equal: since, L^- \neq L^+, the limit L does not exist. Then, x_0 is called a ''jump discontinuity'', ''step discontinuity'', or ''discontinuity of the first kind''. For this type of discontinuity, the function f may have any value at x_0.


Essential discontinuity

For an essential discontinuity, at least one of the two one-sided limits does not exist in \mathbb. (Notice that one or both one-sided limits can be \pm\infty). Consider the function f(x) = \begin \sin\frac & \text x < 1 \\ 0 & \text x = 1 \\ \frac & \text x > 1. \end Then, the point x_0 = 1 is an '. In this example, both L^- and L^+ do not exist in \mathbb, thus satisfying the condition of essential discontinuity. So x_0 is an essential discontinuity, infinite discontinuity, or discontinuity of the second kind. (This is distinct from an
essential singularity In complex analysis, an essential singularity of a function is a "severe" singularity near which the function exhibits odd behavior. The category ''essential singularity'' is a "left-over" or default group of isolated singularities that a ...
, which is often used when studying functions of complex variables). Supposing that f is a function defined on an interval I \subseteq \R, we will denote by D the set of all discontinuities of f on I. By R we will mean the set of all x_0\in I such that f has a ''removable'' discontinuity at x_0. Analogously by J we denote the set constituted by all x_0\in I such that f has a ''jump'' discontinuity at x_0. The set of all x_0\in I such that f has an ''essential'' discontinuity at x_0 will be denoted by E. Of course then D = R \cup J \cup E.


Counting discontinuities of a function

The two following properties of the set D are relevant in the literature. * The set of D is an F_ set. The set of points at which a function is continuous is always a G_ set (see). * If on the interval I, f is monotone then D is at most countable and D = J. This is
Froda's theorem 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 th ...
. Tom Apostol follows partially the classification above by considering only removal and jump discontinuities. His objective is to study the discontinuities of monotone functions, mainly to prove Froda’s theorem. With the same purpose, Walter Rudin and Karl R. Stromberg study also removal and jump discontinuities by using different terminologies. However, furtherly, both authors state that R \cup J is always a countable set (see). The term essential discontinuity seems to have been introduced by John Klippert. Furtherly he also classified essential discontinuities themselves by subdividing the set E into the three following sets: E_1 = \left\, E_2 = \left\, E_3 = \left\. Of course E=E_1 \cup E_2 \cup E_3. Whenever x_0\in E_1, x_0 is called an ''essential discontinuity of first kind''. Any x_0 \in E_2 \cup E_3 is said an ''essential discontinuity of second kind.'' Hence he enlarges the set R \cup J without losing its characteristic of being countable, by stating the following: * The set R \cup J \cup E_2 \cup E_3 is countable.


Rewriting Lebesgue's Theorem

When I= ,b/math> and f is a bounded function, it is well-known of the importance of the set D in the regard of the Riemann integrability of f. In fact, Lebesgue's Theorem (also named Lebesgue-Vitali) theorem) states that f is Riemann integrable on I = ,b/math> if and only if D is a set with Lebesgue's measure zero. In this theorem seems that all type of discontinuities have the same weight on the obstruction that a bounded function f be Riemann integrable on ,b Since countable sets are sets of Lebesgue's measure zero and a countable union of sets with Lebesgue's measure zero is still a set of Lebesgue's mesure zero, we are seeing now that this is not the case. In fact, the discontinuities in the set R \cup J \cup E_2 \cup E_3 are absolutly neutral in the regard of the Riemann integrability of f. The main discontinuities for that purpose are the essential discontinuities of first kind and consequently the Lebesgue-Vitali theorem can be rewritten as follows: * A bounded function, f, is Riemann integrable on ,b/math> if and only if the correspondent set E_1 of all essential discontinuities of first kind of f has Lebesgue's measure zero. The case where E_1 = \varnothing correspond to the following well-known classical complementary situations of Riemann integrability of a bounded function f : , b\to \R: * If f has right-hand limit at each point of , b[ then f is Riemann integrable on [a, b/math> (see) * If f has left-hand limit at each point of ]a, b] then f is Riemann integrable on , b * If f is a regulated function on [a, b] then f is Regulated function#Properties of regulated functions, Riemann integrable on , b


Examples

Thomae's function Thomae's function is a real-valued function of a real variable that can be defined as: f(x) = \begin \frac &\textx = \tfrac\quad (x \text p \in \mathbb Z \text q \in \mathbb N \text\\ 0 &\textx \text \end It is named after Carl Jo ...
is discontinuous at every non-zero
rational point In number theory and algebraic geometry, a rational point of an algebraic variety is a point whose coordinates belong to a given field. If the field is not mentioned, the field of rational numbers is generally understood. If the field is the field ...
, but continuous at every
irrational Irrationality is cognition, thinking, talking, or acting without inclusion of rationality. It is more specifically described as an action or opinion given through inadequate use of reason, or through emotional distress or cognitive deficiency. T ...
point. One easily sees that those discontinuities are all essential of the first kind, that is E_1 = \Q. By the first paragraph, there does not exist a function that is continuous at every
rational Rationality is the quality of being guided by or based on reasons. In this regard, a person acts rationally if they have a good reason for what they do or a belief is rational if it is based on strong evidence. This quality can apply to an abili ...
point, but discontinuous at every irrational point. The
indicator function In mathematics, an indicator function or a characteristic function of a subset of a set is a function that maps elements of the subset to one, and all other elements to zero. That is, if is a subset of some set , one has \mathbf_(x)=1 if x\i ...
of the rationals, also known as the
Dirichlet function In mathematics, the Dirichlet function is the indicator function 1Q or \mathbf_\Q of the set of rational numbers Q, i.e. if ''x'' is a rational number and if ''x'' is not a rational number (i.e. an irrational number). \mathbf 1_\Q(x) = \begin 1 ...
, is discontinuous everywhere. These discontinuities are all essential of the first kind too. Consider now the ternary
Cantor set In mathematics, the Cantor set is a set of points lying on a single line segment that has a number of unintuitive properties. It was discovered in 1874 by Henry John Stephen Smith and introduced by German mathematician Georg Cantor in 1883. Thr ...
\mathcal \subset ,1/math> and its indicator (or characteristic) function \mathbf 1_\mathcal(x) = \begin 1 & x \in \mathcal \\ 0 & x \notin ,1\setminus \mathcal. \end One way to construct the Cantor set \mathcal is given by \mathcal := \bigcap_^\infty C_n where the sets C_n are obtained by recurrence according to C_n = \frac 3 \cup \left(\frac 2 + \frac 3\right) \text n \geq 1, \text C_0 =
, 1 The comma is a punctuation mark that appears in several variants in different languages. It has the same shape as an apostrophe or single closing quotation mark () in many typefaces, but it differs from them in being placed on the baseline (t ...
In view of the discontinuities of the function \mathbf 1_\mathcal(x), let's assume a point x_0\not\in\mathcal. Therefore there exists a set C_n, used in the formulation of \mathcal, which does not contain x_0. That is, x_0 belongs to one of the open intervals which were removed in the construction of C_n. This way, x_0 has a neighbourhood with no points of \mathcal. (In another way, the same conclusion follows taking into account that \mathcal is a closed set and so its complementary with respect to
, 1 The comma is a punctuation mark that appears in several variants in different languages. It has the same shape as an apostrophe or single closing quotation mark () in many typefaces, but it differs from them in being placed on the baseline (t ...
/math> is open). Therefore \mathbf 1_\mathcal only assumes the value zero in some neighbourhood of x_0. Hence \mathbf 1_\mathcal is continuous at x_0. This means that the set D of all discontinuities of \mathbf 1_\mathcal on the interval
, 1 The comma is a punctuation mark that appears in several variants in different languages. It has the same shape as an apostrophe or single closing quotation mark () in many typefaces, but it differs from them in being placed on the baseline (t ...
/math> is a subset of \mathcal. Since \mathcal is a noncountable set with null Lebesgue measure, also D is a null Lebesgue measure set and so in the regard of Lebesgue-Vitali theorem \mathbf 1_\mathcal is a Riemann integrable function. More precisely one has D = \mathcal. In fact, since \mathcal is a rare (closed of empty interior) set, if x_0\in\mathcal then no
neighbourhood A neighbourhood (British English, Irish English, Australian English and Canadian English) or neighborhood (American English; see spelling differences) is a geographically localised community within a larger city, town, suburb or rural are ...
\left(x_0-\varepsilon, x_0+\varepsilon\right) of x_0, can be contained in \mathcal. This way, any neighbourhood of x_0\in\mathcal contains points of \mathcal and points which are not of \mathcal. In terms of the function \mathbf 1_\mathcal this means that both \lim_ \mathbf 1_\mathcal(x) and \lim_ 1_\mathcal(x) do not exist. That is, D = E_1, where by E_1, as before, we denote the set of all essential discontinuities of first kind of the function \mathbf 1_\mathcal. Clearly \int_0^1 \mathbf 1_\mathcal(x)dx = 0.


Discontinuities of derivatives

Let now I \subseteq \R an open interval andf:I\to\mathbb the derivative of a function, F:I\to\mathbb, differentiable on I. That is, F'(x)=f(x) for every x\in I. It is well-known that according to
Darboux's Theorem Darboux's theorem is a theorem in the mathematical field of differential geometry and more specifically differential forms, partially generalizing the Frobenius integration theorem. It is a foundational result in several fields, the chief among ...
the derivative function f: I \to \Reals has the restriction of satisfying the intermediate value property. f can of course be continuous on the interval I. Recall that any continuous function, by Bolzano's Theorem, satisfies the intermediate value property. On the other hand, the intermediate value property does not prevent f from having discontinuities on the interval I. But Darboux's Theorem has an immediate consequence on the type of discontinuities that f can have. In fact, if x_0\in I is a point of discontinuity of f, then necessarily x_0 is an essential discontinuity of f. This means in particular that the following two situations cannot occur: Furtherly, two other situations have to be excluded (see John Klippert): Observe that whenever one of the conditions (i), (ii), (iii), or (iv) is fulfilled for some x_0\in I one can conclude that f fails to possess an antiderivative, F , on the interval I. On the other hand, a new type of discontinuity with respect to any function f:I\to\mathbb can be introduced: an essential discontinuity, x_0 \in I, of the function f, is said to be a ''fundamental essential discontinuity'' of f if \lim_ f(x)\neq\pm\infty and \lim_ f(x)\neq\pm\infty. Therefore if x_0\in I is a discontinuity of a derivative function f:I\to\mathbb, then necessarily x_0 is a fundamental essential discontinuity of f. Notice also that when I= ,b/math> and f:I\to\mathbb is a bounded function, as in the assumptions of Lebesgue's Theorem, we have for all x_0\in (a,b): \lim_ f(x)\neq\pm\infty , \lim_ f(x)\neq\pm\infty, and \lim_ f(x)\neq\pm\infty. Therefore any essential discontinuity of f is a fundamental one.


See also

* * * * ** **


Notes


References


Sources

*


External links

*
"Discontinuity"
by Ed Pegg, Jr.,
The Wolfram Demonstrations Project The Wolfram Demonstrations Project is an organized, open-source collection of small (or medium-size) interactive programs called Demonstrations, which are meant to visually and interactively represent ideas from a range of fields. It is hos ...
, 2007. * * {{SpringerEOM, title=Discontinuity point , id=Discontinuity_point , oldid=12112 , first=L.D. , last=Kudryavtsev, mode=cs1 Theory of continuous functions Mathematical analysis Mathematical classification systems