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Mathematical function with no sudden changes From Wikipedia, the free encyclopedia

In mathematics, a **continuous function** is a function such that a small variation of the argument induces a small variation of the value of the function. 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 to sufficiently small changes of its argument. A **discontinuous function** is a function that is *not continuous*. 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 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 an example, the function *H*(*t*) denoting the height of a growing flower at time t would be considered continuous. In contrast, the function *M*(*t*) denoting the amount of money in a bank account at time t would be considered discontinuous since it "jumps" at each point in time when money is deposited or withdrawn.

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 *x* always produces an infinitely small change of the dependent variable *y* (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 definition and the distinction between pointwise continuity and uniform continuity were first given by Bolzano in the 1830s, but the work wasn't published until the 1930s. Like Bolzano,^{[1]} Karl Weierstrass^{[2]} denied continuity of a function at a point *c* unless it was defined at and on both sides of *c*, but Édouard Goursat^{[3]} allowed the function to be defined only at and on one side of *c*, and Camille Jordan^{[4]} allowed it even if the function was defined only at *c*. All three of those nonequivalent definitions of pointwise continuity are still in use.^{[5]} 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.^{[6]}

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.^{[8]}

Continuity of real functions is usually defined in terms of limits. A function *f* with variable x is *continuous at* the real number c, if the limit of as x tends to c, 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 the interval is contained in the function's domain and the function is continuous at every interval point. A function that is continuous on the interval (the whole real line) is often called simply 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 the interval is contained in the domain of the function, the function is continuous at every interior point of the interval, and the value of the function at each endpoint that belongs to the interval is the limit of the values of the function when the variable tends to the endpoint from the interior of the interval. For example, the function is continuous on its whole domain, which is the closed interval

Many commonly encountered functions are partial functions that have a domain formed by all real numbers, except some isolated points. Examples include the reciprocal function and the tangent function When they are continuous on their domain, one says, in some contexts, that they are continuous, although they are not continuous everywhere. In other contexts, mainly when one is interested in their behavior near the exceptional points, one says they are 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 0, and remain discontinuous whichever value is chosen for defining them at 0. A point where a function is discontinuous is called a *discontinuity*.

Using mathematical notation, several ways exist to define continuous functions in the three senses mentioned above.

Let be a function defined on a subset of the set of real numbers.

This subset is the domain of *f*. Some possible choices include

- : i.e., is the whole set of real numbers. or, for a and b real numbers,
- : is a closed interval, or
- : is an open interval.

In the case of the domain being defined as an open interval, and do not belong to , and the values of and do not matter for continuity on .

The function *f* is *continuous at some point* *c* of its domain if the limit of as *x* approaches *c* through the domain of *f*, exists and is equal to ^{[9]} In mathematical notation, this is written as
In detail this means three conditions: first, *f* has to be defined at *c* (guaranteed by the requirement that *c* is in the domain of *f*). Second, the limit of that equation has to exist. Third, the value of this limit must equal

(Here, we have assumed that the domain of *f* does not have any isolated points.)

A neighborhood of a point *c* is a set that contains, at least, all points within some fixed distance of *c*. Intuitively, a function is continuous at a point *c* if the range of *f* over the neighborhood of *c* shrinks to a single point as the width of the neighborhood around *c* shrinks to zero. More precisely, a function *f* is continuous at a point *c* 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.

One can instead require that for any sequence of points in the domain which converges to *c*, the corresponding sequence converges to In mathematical notation,

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.

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

*C*is non-decreasing