In mathematics, a continuous function is a function such that a small variation of its argument induces at most a small variation of its value. This implies there are no abrupt changes in value, known as discontinuities. More precisely, a function is continuous if arbitrarily small changes in its value can be assured by restricting its argument to sufficiently small changes. A discontinuous function is a function that is . Until the 19th century, mathematicians largely relied on intuitive notions of continuity and considered only continuous functions. The epsilon–delta definition of a limit was introduced to formalize the definition of continuity.
Continuity is one of the core concepts of calculus and mathematical analysis, where arguments and values of functions are real numbers and complex numbers. The concept has been generalized to functions between metric spaces and between topological spaces. The latter are the most general continuous functions, and their definition is the basis of topology.
A stronger form of continuity is uniform continuity. In order theory, especially in domain theory, a related concept of continuity is Scott continuity.
As a practical example, the function denoting the height of a growing flower at time would be considered continuous. In contrast, the function denoting the amount of money in a bank account at time would be considered discontinuous since it "jumps" at each point in time when money is deposited or withdrawn.
History
A form of the epsilon–delta definition of continuity was first given by Bernard Bolzano in 1817. Augustin-Louis Cauchy defined continuity of as follows: an infinitely small increment of the independent variable always produces an infinitely small change of the dependent variable (see e.g. Cours d'Analyse, p. 34). Cauchy defined infinitely small quantities in terms of variable quantities, and his definition of continuity closely parallels the infinitesimal definition used today (see microcontinuity).
The formal definitions for, and distinction between, pointwise continuity and uniform continuity were first given by Bolzano in the 1830s, but the work was not published until the 1930s. Like Bolzano, Karl Weierstrass considered that a function at a point , that is , is continuous if and only if the values of , , and are all defined and equal. Édouard Goursat assumed continuity provided that the function is defined at and is equal to at least one side of the limit , whereas Camille Jordan allowed it even if the function was defined only at . All three of these nonequivalent definitions of pointwise continuity are still in use. Eduard Heine provided the first published definition of uniform continuity in 1872, but based these ideas on lectures given by Peter Gustav Lejeune Dirichlet in 1854.
Real functions Definition
A real function (that is, a function from real numbers to real numbers) can be represented by a graph in the Cartesian plane; such a function is continuous if, roughly speaking, the graph is a single unbroken curve whose domain is the entire real line. A more mathematically rigorous definition is given below.
Continuity of real functions is usually defined in terms of limits. A function with variable is continuous at the real number if the limit of as tends to , is equal to
There are several different definitions of the (global) continuity of a function, which depend on the nature of its domain.
A function is continuous on an open interval if (1) the interval is contained in the function's domain, and (2) the function is continuous at every point in the interval. A function that is continuous on the interval (the whole real line) is often simply called a continuous function; one also says that such a function is continuous everywhere. For example, all polynomial functions are continuous everywhere.
A function is continuous on a semi-open or a closed interval if (1) the interval is contained in the function's domain, (2) the function is continuous at every point in the interval, and (3) the function is continuous at the closed endpoints—that is, technically, the value of the function at a closed endpoint equals the limit of the values of the function as the variable approaches the endpoint from the interior of the interval. For example, the function is continuous on its whole domain, which is the semi-open interval
Many commonly encountered functions are partial functions that have a domain formed by all real numbers, except for some isolated points. Examples include the reciprocal function and the tangent function . When these partial functions are continuous on their domain, they are (in some contexts) said to be continuous, although they are not continuous everywhere. In other contexts, mainly when one is interested in their behavior near the exceptional points, they are said to be discontinuous.
A partial function is discontinuous at a point if the point belongs to the topological closure of its domain, and either the point does not belong to the domain of the function or the function is not continuous at the point. For example, the functions and are discontinuous at , and remain discontinuous whichever value is chosen for defining them at . A point where a function is discontinuous is called a discontinuity. At a point where a function is not defined, and therefore discontinuous, the discontinuity is removable if a value of the function can be chosen at that point to make the function continuous. For example, the function has a removable discontinuity at zero, since , and the discontinuity at 0 of is not removable, since does not exist.
Using mathematical notation, several ways exist to define continuous functions in the three senses mentioned above.
Let be a function whose domain is a subset of the real numbers
Some (but not all) possibilities for are: is the whole real line; that is, is a closed interval of the form where and are real numbers is an open interval of the form where and are real numbers
In the case of an open interval, and do not belong to , and the values and are not defined, and if they are, they do not matter for continuity on .
Definition in terms of limits of functions The function is continuous at some point of its domain if the limit of as x approaches c through the domain of , exists and is equal to , section II.4 In mathematical notation, this is written as
In detail this means three conditions: first, has to be defined at (guaranteed by the requirement that be in the domain of ). Second, the limit of that equation must exist. Third, the value of this limit must equal
(Here, we have assumed that the domain of does not have any isolated points.)
Definition in terms of neighborhoods A neighborhood of a point is a set that contains, at least, all points within some fixed distance of . Intuitively, a function is continuous at a point if the range of over the neighborhood of shrinks to a single point as the width of the neighborhood around shrinks to zero. More precisely, a function is continuous at a point of its domain if, for any neighborhood there is a neighborhood in its domain such that whenever
As neighborhoods are defined in any topological space, this definition of a continuous function applies not only for real functions but also when the domain and the codomain are topological spaces and is thus the most general definition. It follows that a function is automatically continuous at every isolated point of its domain. For example, every real-valued function on the integers is continuous.
Definition in terms of limits of sequences
One can instead require that for any sequence of points in the domain which converges to , the corresponding sequence converges to In mathematical notation,
Weierstrass and Jordan definitions (epsilon–delta) of continuous functions
Explicitly including the definition of the limit of a function, we obtain a self-contained definition: Given a function as above and an element of the domain , is said to be continuous at the point when the following holds: For any positive real number however small, there exists some positive real number such that for all in the domain of with the value of satisfies
Alternatively written, continuity of at means that for every there exists a such that for all :
More intuitively, we can say that if we want to get all the values to stay in some small neighborhood around we need to choose a small enough neighborhood for the values around If we can do that no matter how small the neighborhood is, then is continuous at
In modern terms, this is generalized by the definition of continuity of a function with respect to a basis for the topology, here the metric topology.
Weierstrass had required that the interval be entirely within the domain , but Jordan removed that restriction.
Definition in terms of control of the remainder In proofs and numerical analysis, we often need to know how fast limits are converging, or in other words, control of the remainder. We can formalize this to a definition of continuity. A function is called a control function if is non-decreasing and .
A function is C-continuous at if there exists a neighborhood such that
A function is continuous in if it is for some control function .
This approach leads naturally to refining the notion of continuity by restricting the set of admissible control functions. For a given set of control functions , a function is if it is for some For example, the Lipschitz, the Hölder continuous functions of exponent and the uniformly continuous functions below are defined by the set of control functions
respectively.
Definition using oscillation
Continuity can also be defined in terms of oscillation: a function is continuous at a point if and only if its oscillation at that point is zero;Introduction to Real Analysis, updated April 2010, William F. Trench, Theorem 3.5.2, p. 172 in symbols, A benefit of this definition is that it discontinuity: the oscillation gives how the function is discontinuous at a point.
This definition is helpful in descriptive set theory to study the set of discontinuities and continuous points—the continuous points are the intersection of the sets where the oscillation is less than (hence a set)—and gives a rapid proof of one direction of the Lebesgue integrability condition.Introduction to Real Analysis, updated April 2010, William F. Trench, 3.5 "A More Advanced Look at the Existence of the Proper Riemann Integral", pp. 171–177
The oscillation is equivalent to the definition by a simple re-arrangement and by using a limit (lim sup, lim inf) to define oscillation: if (at a given point) for a given there is no that satisfies the definition, then the oscillation is at least and conversely if for every there is a desired the oscillation is 0. The oscillation definition can be naturally generalized to maps from a topological space to a metric space.
Definition using the hyperreals Cauchy defined the continuity of a function in the following intuitive terms: an infinitesimal change in the independent variable corresponds to an infinitesimal change of the dependent variable (see Cours d'analyse, page 34). Non-standard analysis is a way of making this mathematically rigorous. The real line is augmented by adding infinite and infinitesimal numbers to form the hyperreal numbers. In nonstandard analysis, continuity can be defined as follows.
(see microcontinuity). In other words, an infinitesimal increment of the independent variable always produces an infinitesimal change of the dependent variable, giving a modern expression to Augustin-Louis Cauchy's definition of continuity.
Rules for continuity
Proving the continuity of a function by a direct application of the definition is generally not an easy task. Fortunately, in practice, most functions are built from simpler functions, and their continuity can be deduced immediately from the way they are defined, by applying the following rules: