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continuous function
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{{Calculus}}In mathematics, a continuous function is a function for which sufficiently small changes in the input result in arbitrarily small changes in the output. Otherwise, a function is said to be a discontinuous function. A continuous function with a continuous inverse function is called a homeomorphism.Continuity of functions is one of the core concepts of topology, which is treated in full generality below. The introductory portion of this article focuses on the special case where the inputs and outputs of functions are real numbers. A stronger form of continuity is uniform continuity. In addition, this article discusses the definition for the more general case of functions between two metric spaces. In order theory, especially in domain theory, one considers a notion of continuity known as Scott continuity. Other forms of continuity do exist but they are not discussed in this article.As an example, consider the function h(t), which describes the height of a growing flower at time t. This function is continuous. By contrast, if M(t) denotes the amount of money in a bank account at time t, then the function jumps at each point in time when money is deposited or withdrawn, so the function M(t) is discontinuous.

History

A form of the epsilon–delta definition of continuity was first given by Bernard Bolzano in 1817. Augustin-Louis Cauchy defined continuity of y=f(x) as follows: an infinitely small increment alpha of the independent variable x always produces an infinitely small change f(x+alpha)-f(x) 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,{{citation|last1=Bolzano|first1=Bernard|title=Rein analytischer Beweis des Lehrsatzes dass zwischen je zwey Werthen, die ein entgegengesetztes Resultat gewaehren, wenigstens eine reele Wurzel der Gleichung liege|publisher=Haase|location=Prague|date=1817}} Karl Weierstrass{{Citation | last1=Dugac | first1=Pierre | title=Eléments d'Analyse de Karl Weierstrass | journal=Archive for History of Exact Sciences | year=1973 | volume=10 | pages=41–176 | doi=10.1007/bf00343406}} denied continuity of a function at a point c unless it was defined at and on both sides of c, but Édouard Goursat{{Citation | last1=Goursat | first1=E. | title=A course in mathematical analysis | publisher=Ginn | location=Boston | year=1904 | page=2}} allowed the function to be defined only at and on one side of c, and Camille Jordan{{Citation | last1=Jordan | first1=M.C. | title=Cours d'analyse de l'École polytechnique | publisher=Gauthier-Villars | location=Paris | edition=2nd |year=1893 | volume=1|page=46}} allowed it even if the function was defined only at c. All three of those nonequivalent definitions of pointwise continuity are still in use.{{Citation|last1=Harper|first1=J.F.|title=Defining continuity of real functions of real variables|journal=BSHM Bulletin: Journal of the British Society for the History of Mathematics|year=2016|doi=10.1080/17498430.2015.1116053|pages=1–16}} 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.{{citation|last1=Rusnock|first1=P.|last2=Kerr-Lawson|first2=A.|title=Bolzano and uniform continuity|journal=Historia Mathematica|volume=32|year=2005|pages=303–311|issue=3|doi=10.1016/j.hm.2004.11.003}}

Real functions

Definition

(File:Function-1 x.svg|thumb|The function f(x)=tfrac 1x is continuous on the domain Rsmallsetminus {0}, but is not continuous over the domain R because it is undefined at x=0)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.WEB,weblink Continuity and Discontinuity, Speck, Jared, 2014, 3, 2016-09-02, MIT Math, Example 5. The function 1/x is continuous on (0, ∞) and on (−∞, 0), i.e., for x > 0 and for x < 0, in other words, at every point in its domain. However, it is not a continuous function since its domain is not an interval. It has a single point of discontinuity, namely x = 0, and it has an infinite discontinuity there., A rigorous definition of continuity of real functions is usually given in a first course in calculus in terms of the idea of a limit. First, a function {{math|f}} with variable {{mvar|x}} is said to be continuous at the point {{math|c}} on the real line, if the limit of {{math|f(x)}}, as {{mvar|x}} approaches that point {{math|c}}, is equal to the value {{math|f(c)}}; and second, the function (as a whole) is said to be continuous, if it is continuous at every point. A function is said to be discontinuous (or to have a discontinuity) at some point when it is not continuous there. These points themselves are also addressed as discontinuities.There are several different definitions of continuity of a function. Sometimes a function is said to be continuous if it is continuous at every point in its domain. In this case, the function {{math|f(x) {{=}} tan(x)}}, with the domain of all real {{math|x ≠ (2n+1)Ï€/2}}, {{math|n}} any integer, is continuous. Sometimes an exception is made for boundaries of the domain. For example, the graph of the function {{math|f(x) {{=}} {{sqrt|x}}}}, with the domain of all non-negative reals, has a left-hand endpoint. In this case only the limit from the right is required to equal the value of the function. Under this definition f is continuous at the boundary {{math|x {{=}} 0}} and so for all non-negative arguments. The most common and restrictive definition is that a function is continuous if it is continuous at all real numbers. In this case, the previous two examples are not continuous, but every polynomial function is continuous, as are the sine, cosine, and exponential functions. Care should be exercised in using the word continuous, so that it is clear from the context which meaning of the word is intended.Using mathematical notation, there are several ways to define continuous functions in each of the three senses mentioned above.Let
fcolon D rightarrow mathbf R quad be a function defined on a subset D of the set mathbf R of real numbers.
This subset D is the domain of f. Some possible choices include
D = mathbf R quad ( D is the whole set of real numbers), or, for a and b real numbers, D = [a, b] = {x in mathbf R ,|, a leq x leq b } quad ( D is a closed interval), or D = (a, b) = {x in mathbf R ,|, a < x < b } quad ( D is an open interval).
In case of the domain D being defined as an open interval, a and b are no boundaries in the above sense and the values of f(a) and f(b) do not matter for continuity on D.

Definition in terms of limits of functions

The function f is continuous at some point c of its domain if the limit of f(x), as x approaches c through the domain of f, exists and is equal to f(c).{{Citation | last1=Lang | first1=Serge | author1-link=Serge Lang | title=Undergraduate analysis | publisher=Springer-Verlag | location=Berlin, New York | edition=2nd | series=Undergraduate Texts in Mathematics | isbn=978-0-387-94841-6 | year=1997}}, section II.4 In mathematical notation, this is written as
lim_{x to c}{f(x)} = f(c).
In detail this means three conditions: first, f has to be defined at c. Second, the limit on the left hand side of that equation has to exist. Third, the value of this limit must equal f(c).(We have here assumed that the domain of f does not have any isolated points. For example, an interval or union of intervals has no isolated points.)

Definition in terms of neighborhoods

A neighborhood of a point c is a set that contains all points of the domain within some fixed distance of c. Intuitively, a function is continuous at a point c if the range of the restriction of f to a neighborhood of c shrinks to a single point f(c) 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 N_1(f(c)) there is a neighborhood N_2(c) such that f(x)in N_1(f(c)) whenever xin N_2(c).This definition only requires that the domain and the codomain are topological spaces and is thus the most general definition. It follows from this definition that a function f is automatically continuous at every isolated point of its domain. As a specific example, every real valued function on the set of integers is continuous.

Definition in terms of limits of sequences

(File:Continuity of the Exponential at 0.svg|thumb|The sequence exp(1/n) converges to exp(0))One can instead require that for any sequence (x_n)_{ninmathbb{N}} of points in the domain which converges to c, the corresponding sequence left(f(x_n)right)_{nin mathbb{N}} converges to f(c). In mathematical notation, forall (x_n)_{ninmathbb{N}} subset D:lim_{ntoinfty} x_n=c Rightarrow lim_{ntoinfty} f(x_n)=f(c),.

Weierstrass and Jordan definitions (epsilon–delta) of continuous functions

(File:Example of continuous function.svg|right|thumb|Illustration of the ε-δ-definition: for ε=0.5, c=2, the value δ=0.5 satisfies the condition of the definition.)Explicitly including the definition of the limit of a function, we obtain a self-contained definition:Given a function f : D â†’ R as above and an element x0 of the domain D, f is said to be continuous at the point x0 when the following holds: For any number ε > 0, however small, there exists some number δ > 0 such that for all x in the domain of f with x0 âˆ’ Î´  0 such that for all x âˆˆ D :
| x - x_0 | < delta Rightarrow | f(x) - f(x_0) | < varepsilon.
More intuitively, we can say that if we want to get all the f(x) values to stay in some small neighborhood around f(x0), we simply need to choose a small enough neighborhood for the x values around x0. If we can do that no matter how small the f(x) neighborhood is, then f is continuous at x0.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 x0 âˆ’ Î´  0} respectively
mathcal{C}_{text{Hölder}-alpha} = {C | C(delta) = K |delta|^alpha, K > 0} .

Definition using oscillation

File:Rapid Oscillation.svg|thumb|The failure of a function to be continuous at a point is quantified by its oscillation.]]Continuity can also be defined in terms of oscillation: a function f is continuous at a point x0 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, omega_f(x_0) = 0. A benefit of this definition is that it quantifies discontinuity: the oscillation gives how much the function is discontinuous at a point.This definition is useful 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 ε (hence a Gδ set) – and gives a very quick 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. 171–177The oscillation is equivalent to the ε-δ 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 ε0 there is no δ that satisfies the ε-δ definition, then the oscillation is at least ε0, and conversely if for every ε there is a desired δ, 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 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 the addition of infinite and infinitesimal numbers to form the hyperreal numbers. In nonstandard analysis, continuity can be defined as follows.
A real-valued function f is continuous at x if its natural extension to the hyperreals has the property that for all infinitesimal dx, {{nowrap|f(x+dx) − f(x)}} is infinitesimalWEB,weblink Elementary Calculus, wisc.edu,
(see microcontinuity). In other words, an infinitesimal increment of the independent variable always produces to an infinitesimal change of the dependent variable, giving a modern expression to Augustin-Louis Cauchy's definition of continuity.

Construction of continuous functions

File:Brent method example.svg|right|thumb|The graph of a cubic functioncubic functionChecking the continuity of a given function can be simplified by checking one of the above defining properties for the building blocks of the given function. It is straightforward to show that the sum of two functions, continuous on some domain, is also continuous on this domain. Given
f, gcolon D rightarrow mathbf R,
then the sum of continuous functions
s = f + g, defined by s(x) = f(x) + g(x) for all xin D
is continuous in D.The same holds for the product of continuous functions.
p = f cdot g, defined by p(x) = f(x) cdot g(x) for all x in D,
is continuous in D.Combining the above preservations of continuity and the continuity of constant functions and of the identity function I(x) = x {{nowrap|on mathbf R,}} one arrives at the continuity of all polynomial functions {{nowrap|on mathbf R,}} such as
{{math|1=f(x) = x3 + x2 - 5x + 3}}
(pictured, above).File:Homografia.svg|right|thumb|The graph of a continuous rational function. The function is not defined for x=−2. The vertical and horizontal lines are asymptoteasymptoteIn the same way it can be shown that the reciprocal of a continuous function
r = 1/f, defined by r(x) = 1/f(x) for all x in D such that f(x) ne 0,
is continuous in Dsmallsetminus {x:f(x) = 0}.This implies that, excluding the roots of g, the quotient of continuous functions
q = f/g, defined by g(x) = f(x)/g(x) for all x in D, such that g(x) ne 0,
is also continuous on Dsmallsetminus {x:g(x) = 0}.For example, the function (pictured)
y(x) = frac {2x-1} {x+2}
is defined for all real numbers {{nowrap|x ≠ −2}} and is continuous at every such point. Thus it is a continuous function. The question of continuity at {{nowrap|x {{=}} −2}} does not arise, since {{nowrap|x {{=}} −2}} is not in the domain of y. There is no continuous function F: R → R that agrees with y(x) for all {{nowrap|x ≠ −2}}.(File:Si cos.svg|thumb|The sinc and the cos functions)Since the sine-function is continuous on all reals, the sinc function G(x) = sin x/x, when defined for all real x ≠ 0, is continuous at these points. Thus G is a continuous function there, too. However, unlike the previous example, G can be extended to a continuous function on all real numbers, by defining the value G(0) to be the limit of G(x), when x approaches 0, i.e.,
G(0) = lim_{xrightarrow 0}frac{sin x}{x} = 1
and thus
G(x) = begin{cases}frac {sin (x)}x & text{ if }x ne 01 & text{ if }x = 0,end{cases}the sinc-function becomes a continuous function on all real numbers. The term removable singularity is used in such cases, when (re)defining values of a function to coincide with the appropriate limits make a function continuous at specific points.A more involved construction of continuous functions is the function composition. Given two continuous functions
fcolon D_f ;(subseteq mathbf R) rightarrow R_f; (subseteq mathbf R)quad and quad gcolon D_g; (R_f subseteq D_g) rightarrow R_g ;(subseteq mathbf R),
their composition being denoted as
c = g circ f colon D_f rightarrow mathbf R, and defined by c(x) = g(f(x)), is continuous.
This construction allows stating, for example, that
e^{sin(ln x)} is continuous for all x > 0.

Examples of discontinuous functions

(File:Discontinuity of the sign function at 0.svg|thumb|300px|Plot of the signum function. It shows that lim_{ntoinfty} sgnleft(tfrac 1nright) neqsgnleft(lim_{ntoinfty} tfrac 1nright). Thus, the signum function is discontinuous at 0 for point 2.1.3.)An example of a discontinuous function is the Heaviside step function H, defined by
H(x) = begin{cases}
1 & text{ if } x ge 0end{cases}Pick for instance varepsilon = 1/2. Then there is no {{nowrap|delta-neighborhood}} around x = 0, i.e. no open interval (-delta,;delta) with delta > 0, that will force all the H(x) values to be within the {{nowrap|varepsilon-neighborhood}} of H(0), i.e. within (1/2,;3/2). Intuitively we can think of this type of discontinuity as a sudden jump in function values.Similarly, the signum or sign function
sgn(x) = begin{cases}
1 & text{ if }x > 0
0 & text{ if }x = 0
-1 & text{ if }x < 0end{cases}is discontinuous at x = 0 but continuous everywhere else. Yet another example: the function
f(x)=begin{cases}
sinleft(x^{-2}right)&text{ if }x ne 0
0&text{ if }x = 0
end{cases}is continuous everywhere apart from x = 0.(File:Thomae function (0,1).svg|200px|right|thumb|Point plot of Thomae's function on the interval (0,1). The topmost point in the middle shows f(1/2) = 1/2.)Besides plausible continuities and discontinuities like above, there are also functions with a behavior, often coined pathological, for example, Thomae's function,
f(x)=begin{cases}
1 &text{ if }x=0frac{1}{q}&text{ if }x=frac{p}{q}text{(in lowest terms) is a rational number}
0&text{ if }xtext{ is irrational}.
end{cases}is continuous at all irrational numbers and discontinuous at all rational numbers. In a similar vein, Dirichlet's function, the indicator function for the set of rational numbers,
D(x)=begin{cases}
0&text{ if }xtext{ is irrational } (in mathbb{R} smallsetminus mathbb{Q})
1&text{ if }xtext{ is rational } (in mathbb{Q})
end{cases}is nowhere continuous.

Properties

Intermediate value theorem

The intermediate value theorem is an existence theorem, based on the real number property of completeness, and states:
If the real-valued function f is continuous on the closed interval [ab] and k is some number between f(a) and f(b), then there is some number c in [ab] such that f(c) = k.
For example, if a child grows from 1 m to 1.5 m between the ages of two and six years, then, at some time between two and six years of age, the child's height must have been 1.25 m.As a consequence, if f is continuous on [ab] and f(a) and f(b) differ in sign, then, at some point c in [ab], f(c) must equal zero.

Extreme value theorem

The extreme value theorem states that if a function f is defined on a closed interval [a,b] (or any closed and bounded set) and is continuous there, then the function attains its maximum, i.e. there exists c âˆˆ [a,b] with f(c) ≥ f(x) for all x âˆˆ [a,b]. The same is true of the minimum of f. These statements are not, in general, true if the function is defined on an open interval (a,b) (or any set that is not both closed and bounded), as, for example, the continuous function f(x) = 1/x, defined on the open interval (0,1), does not attain a maximum, being unbounded above.

Relation to differentiability and integrability

Every differentiable function
fcolon (a, b) rightarrow mathbf R
is continuous, as can be shown. The converse does not hold: for example, the absolute value function
f(x)=|x| = begin{cases}
;; x & text{ if }x geq 0
-x & text{ if }x < 0
end{cases}is everywhere continuous. However, it is not differentiable at x = 0 (but is so everywhere else). Weierstrass's function is also everywhere continuous but nowhere differentiable.The derivative f′(x) of a differentiable function f(x) need not be continuous. If f′(x) is continuous, f(x) is said to be continuously differentiable. The set of such functions is denoted C1({{open-open|a, b}}). More generally, the set of functions
fcolon Omega rightarrow mathbf R
(from an open interval (or open subset of R) Ω to the reals) such that f is n times differentiable and such that the n-th derivative of f is continuous is denoted Cn(Ω). See differentiability class. In the field of computer graphics, these three levels are sometimes called G0 (continuity of position), G1 (continuity of tangency), and G2 (continuity of curvature).Every continuous function
fcolon [a, b] rightarrow mathbf R
is integrable (for example in the sense of the Riemann integral). The converse does not hold, as the (integrable, but discontinuous) sign function shows.

Pointwise and uniform limits

(File:Uniform continuity animation.gif|A sequence of continuous functions fn(x) whose (pointwise) limit function f(x) is discontinuous. The convergence is not uniform.|right|thumb)Given a sequence
f_1, f_2, dotsc colon I rightarrow mathbf R
of functions such that the limit
f(x) := lim_{n rightarrow infty} f_n(x)
exists for all x in D, the resulting function f(x) is referred to as the pointwise limit of the sequence of functions (f'n)n∈N. The pointwise limit function need not be continuous, even if all functions f'n are continuous, as the animation at the right shows. However, f is continuous if all functions fn are continuous and the sequence converges uniformly, by the uniform convergence theorem. This theorem can be used to show that the exponential functions, logarithms, square root function, trigonometric functions are continuous.

Directional and semi-continuity

Image:Right-continuous.svg|A right-continuous functionImage:Left-continuous.svg|A left-continuous functionDiscontinuous functions may be discontinuous in a restricted way, giving rise to the concept of directional continuity (or right and left continuous functions) and semi-continuity. Roughly speaking, a function is right-continuous if no jump occurs when the limit point is approached from the right. Formally, f is said to be right-continuous at the point c if the following holds: For any number ε > 0 however small, there exists some number δ > 0 such that for all x in the domain with {{nowrap|c < x < c + δ}}, the value of f(x) will satisfy
|f(x) - f(c)| < varepsilon.
This is the same condition as for continuous functions, except that it is required to hold for x strictly larger than c only. Requiring it instead for all x with {{nowrap|c − δ < x < c}} yields the notion of left-continuous functions. A function is continuous if and only if it is both right-continuous and left-continuous.A function f is lower semi-continuous if, roughly, any jumps that might occur only go down, but not up. That is, for any ε > 0, there exists some number δ > 0 such that for all x in the domain with {{nowrap|{{abs|x − c}} < δ}}, the value of f(x) satisfies
f(x) geq f(c) - epsilon.
The reverse condition is upper semi-continuity.

Continuous functions between metric spaces

The concept of continuous real-valued functions can be generalized to functions between metric spaces. A metric space is a set X equipped with a function (called metric) dX, that can be thought of as a measurement of the distance of any two elements in X. Formally, the metric is a function
d_X colon X times X rightarrow mathbf R
that satisfies a number of requirements, notably the triangle inequality. Given two metric spaces (X, dX) and (Y, dY) and a function
fcolon X rightarrow Y
then f is continuous at the point c in X (with respect to the given metrics) if for any positive real number ε, there exists a positive real number δ such that all x in X satisfying dX(x, c) < δ will also satisfy dY(f(x), f(c)) < ε. As in the case of real functions above, this is equivalent to the condition that for every sequence (x'n) in X with limit lim x'n = c, we have lim f(x'n) = f(c). The latter condition can be weakened as follows: f is continuous at the point c if and only if for every convergent sequence (x'n) in X with limit c, the sequence (f(xn)) is a Cauchy sequence, and c is in the domain of f.The set of points at which a function between metric spaces is continuous is a Gδ set â€“ this follows from the ε-δ definition of continuity.This notion of continuity is applied, for example, in functional analysis. A key statement in this area says that a linear operator
Tcolon V rightarrow W
between normed vector spaces V and W (which are vector spaces equipped with a compatible norm, denoted ||x||)is continuous if and only if it is bounded, that is, there is a constant K such that
|T(x)| leq K |x|
for all x in V.

Uniform, Hölder and Lipschitz continuity

(File:Lipschitz continuity.png|thumb|For a Lipschitz continuous function, there is a double cone (shown in white) whose vertex can be translated along the graph, so that the graph always remains entirely outside the cone.)The concept of continuity for functions between metric spaces can be strengthened in various ways by limiting the way δ depends on ε and c in the definition above. Intuitively, a function f as above is uniformly continuous if the δ doesnot depend on the point c. More precisely, it is required that for every real number ε > 0 there exists δ > 0 such that for every cb âˆˆ X with dX(bc

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