Continuous Functions - Revision history

Khanh at 05:56, 6 November 2021

2021-11-06T05:56:38Z

Khanh: /* Continuity at a Point */

2021-10-21T03:36:54Z

Continuity at a Point

Khanh: /* Continuity at a Point */

2021-10-21T03:34:00Z

Continuity at a Point

Khanh: /* Topological Continuity */

2021-10-21T03:26:51Z

Topological Continuity

Lila: /* Operations */

2021-10-20T18:09:14Z

Operations

Lila at 17:33, 20 October 2021

2021-10-20T17:33:03Z

Lila at 17:31, 20 October 2021

2021-10-20T17:31:56Z

Lila: /* The Intermediate Value Theorem */

2021-10-20T17:08:53Z

The Intermediate Value Theorem

Lila: /* Proof */

2021-10-14T20:58:26Z

Proof

Lila: /* Extreme Value Theorem */

2021-10-14T20:57:42Z

Extreme Value Theorem

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 It must be mentioned here that the term "Open Set" can be defined in much more general settings than the set of reals or even metric spaces, and hence the utility of this characterization.
-==Resources==
+== Licensing ==
-* [https://en.wikibooks.org/wiki/Real_Analysis/Continuity Continuity], Wikibooks: Real Analysis
+Content obtained and/or adapted from:

@@ Line 10: / Line 10: @@
 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),</math> as {{mvar|x}} approaches that point {{math|c}}, is equal to the value <math>f(c)</math>; 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),</math> with the domain of all real <math>x \neq \pi \frac{2 n + 1}{2},</math> <math>n</math> 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},</math> 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</math> 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.
+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),</math> with the domain of all real <math>x \neq \pi \frac{2 n + 1}{2},</math> <math>n</math> 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},</math> 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</math> 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.
@@ Line 38: / Line 38: @@
 ====Definition in terms of limits of sequences====
 [[File:Continuity of the Exponential at 0.svg|thumb|The sequence <math>\exp (1/n)</math> converges to exp(0)]]
-One can instead require that for any [[sequence (mathematics)|sequence]] <math>(x_n)_{n \in \N}</math> of points in the domain which converges to ''c'', the corresponding sequence <math>\left(f(x_n)\right)_{n\in \N}</math> converges to <math>f(c).</math>  In mathematical notation, <math>\forall (x_n)_{n \in \N} \subset D:\lim_{n\to\infty} x_n = c \Rightarrow \lim_{n\to\infty} f(x_n) = f(c)\,.</math>
+One can instead require that for any sequence <math>(x_n)_{n \in \N}</math> of points in the domain which converges to ''c'', the corresponding sequence <math>\left(f(x_n)\right)_{n\in \N}</math> converges to <math>f(c).</math>  In mathematical notation, <math>\forall (x_n)_{n \in \N} \subset D:\lim_{n\to\infty} x_n = c \Rightarrow \lim_{n\to\infty} f(x_n) = f(c)\,.</math>
 ====Weierstrass and Jordan definitions (epsilon–delta) of continuous functions====

@@ Line 6: / Line 6: @@
 [[File:Function-1 x.svg|thumb|The function <math>f(x)=\tfrac 1 x</math> is continuous on the domain <math>\R\setminus \{0\}</math>, but is not continuous over the domain <math>\R</math> because it is undefined at <math>x=0</math>]]
-A [[real function]], that is a [[function (mathematics)|function]] from [[real number]]s to real numbers, can be represented by a [[graph of a function|graph]] in the [[Cartesian coordinate system|Cartesian plane]]; such a function is continuous if, roughly speaking, the graph is a single unbroken [[curve]] whose [[domain of a function|domain]] is the entire real line.  A more mathematically rigorous definition is given below.<ref>{{cite web | url=http://math.mit.edu/~jspeck/18.01_Fall%202014/Supplementary%20notes/01c.pdf | title=Continuity and Discontinuity | last1=Speck | first1=Jared | year=2014 | page=3 | access-date=2016-09-02 | website=MIT Math | quote=Example 5. The function <math>1/x</math> is continuous on <math>(0, \infty)</math> and on <math>(-\infty, 0),</math> i.e., for <math>x > 0</math> and for <math>x < 0,</math> 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 <math>x = 0,</math> and it has an infinite discontinuity there.}}</ref>
+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.
-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 (mathematics)|limit]]. First, a function {{math|''f''}} with variable {{mvar|x}} is said to be continuous {{em|at the point}} {{math|c}} on the real line, if the limit of <math>f(x),</math> as {{mvar|x}} approaches that point {{math|c}}, is equal to the value <math>f(c)</math>; and second, the ''function (as a whole)'' is said to be {{em|continuous}}, if it is continuous at every point. A function is said to be {{em|discontinuous}} (or to have a ''discontinuity'') at some point when it is not continuous there. These points themselves are also addressed as {{em|discontinuities}}.
+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),</math> as {{mvar|x}} approaches that point {{math|c}}, is equal to the value <math>f(c)</math>; 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),</math> with the domain of all real <math>x \neq \pi \frac{2 n + 1}{2},</math> <math>n</math> 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},</math> with the domain of all non-negative reals, has a {{em|left-hand}} endpoint. In this case only the limit from the {{em|right}} is required to equal the value of the function. Under this definition ''f'' is continuous at the boundary <math>x = 0</math> 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 {{em|continuous}}, so that it is clear from the context which meaning of the word is intended.
+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),</math> with the domain of all real <math>x \neq \pi \frac{2 n + 1}{2},</math> <math>n</math> 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},</math> 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</math> 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
-:<math>f : D \to \R \quad</math> be a function defined on a [[subset]] <math> D </math> of the set <math>\R</math> of real numbers.
+:<math>f : D \to \R \quad</math> be a function defined on a subset <math> D </math> of the set <math>\R</math> of real numbers.
-This subset <math> D </math> is the [[Domain of a function|domain]] of ''f''. Some possible choices include
+This subset <math> D </math> is the domain of ''f''. Some possible choices include
 :<math>D = \R \quad </math> (<math> D </math> is the whole set of real numbers), or, for ''a'' and ''b'' real numbers,
-:<math>D = [a, b] = \{x \in \R \mid a \leq x \leq b \} \quad </math> (<math> D </math> is a [[closed interval]]), or
+:<math>D = [a, b] = \{x \in \R \mid a \leq x \leq b \} \quad </math> (<math> D </math> is a closed interval), or
-:<math>D = (a, b) = \{x \in \R \mid a < x < b \} \quad </math> (<math> D </math> is an [[open interval]]).
+:<math>D = (a, b) = \{x \in \R \mid a < x < b \} \quad </math> (<math> D </math> is an open interval).
 In case of the domain <math>D</math> being defined as an open interval, <math>a</math> and <math>b</math> do not belong to <math>D</math>, and the values of <math>f(a)</math> and <math>f(b)</math> do not matter for continuity on <math>D</math>.
 ====Definition in terms of limits of functions====
-The function ''f'' is ''continuous at some point'' ''c'' of its domain if the [[limit of a function|limit]] of <math>f(x),</math> as ''x'' approaches ''c'' through the domain of ''f'',  exists and is equal to <math>f(c).</math><ref>{{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</ref> In mathematical notation, this is written as
+The function ''f'' is ''continuous at some point'' ''c'' of its domain if the limit of <math>f(x),</math> as ''x'' approaches ''c'' through the domain of ''f'',  exists and is equal to <math>f(c).</math> In mathematical notation, this is written as
 <math display="block">\lim_{x \to c}{f(x)} = f(c).</math>
 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 on the left hand side of that equation has to exist.  Third, the value of this limit must equal <math>f(c).</math>
-(Here, we have assumed that the domain of ''f'' does not have any [[isolated point]]s.)
+(Here, we have assumed that the domain of ''f'' does not have any isolated points.)
 ====Definition in terms of neighborhoods====
-A [[neighborhood (mathematics)|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 <math>f(c)</math> 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 <math>N_1(f(c))</math> there is a neighborhood <math>N_2(c)</math> in its domain such that <math>f(x) \in N_1(f(c))</math> whenever <math>x\in N_2(c).</math>
+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 <math>f(c)</math> 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 <math>N_1(f(c))</math> there is a neighborhood <math>N_2(c)</math> in its domain such that <math>f(x) \in N_1(f(c))</math> whenever <math>x\in N_2(c).</math>
-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.
+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 <math>\exp (1/n)</math> converges to exp(0)]]
-One can instead require that for any [[sequence (mathematics)|sequence]] <math>(x_n)_{n \in \N}</math> of points in the domain which [[convergent sequence|converges]] to ''c'', the corresponding sequence <math>\left(f(x_n)\right)_{n\in \N}</math> converges to <math>f(c).</math>  In mathematical notation, <math>\forall (x_n)_{n \in \N} \subset D:\lim_{n\to\infty} x_n = c \Rightarrow \lim_{n\to\infty} f(x_n) = f(c)\,.</math>
+One can instead require that for any [[sequence (mathematics)|sequence]] <math>(x_n)_{n \in \N}</math> of points in the domain which converges to ''c'', the corresponding sequence <math>\left(f(x_n)\right)_{n\in \N}</math> converges to <math>f(c).</math>  In mathematical notation, <math>\forall (x_n)_{n \in \N} \subset D:\lim_{n\to\infty} x_n = c \Rightarrow \lim_{n\to\infty} f(x_n) = f(c)\,.</math>
 ====Weierstrass and Jordan definitions (epsilon–delta) of continuous functions====
@@ Line 49: / Line 49: @@
 <math display="block">\left|x - x_0\right| < \delta \text{ implies } |f(x) - f(x_0)| < \varepsilon.</math>
-More intuitively, we can say that if we want to get all the <math>f(x)</math> values to stay in some small [[Topological neighbourhood|neighborhood]] around <math>f\left(x_0\right),</math> we simply need to choose a small enough neighborhood for the ''x'' values around <math>x_0.</math> If we can do that no matter how small the <math>f(x)</math> neighborhood is, then ''f'' is continuous at <math>x_0.</math>
+More intuitively, we can say that if we want to get all the <math>f(x)</math> values to stay in some small neighborhood around <math>f\left(x_0\right),</math> we simply need to choose a small enough neighborhood for the ''x'' values around <math>x_0.</math> If we can do that no matter how small the <math>f(x)</math> neighborhood is, then ''f'' is continuous at <math>x_0.</math>
-In modern terms, this is generalized by the definition of continuity of a function with respect to a [[basis (topology)|basis for the topology]], here the [[metric topology]].
+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 <math>x_0 - \delta < x < x_0 + \delta</math> be entirely within the domain ''D'', but Jordan removed that restriction.
 ==Continuity on an Interval==

@@ Line 217: / Line 217: @@
 ===Topological Continuity===
-As mentioned, the idea of continuous functions is used in several areas of mathematics, most notably in [[Topology]]. A different characterization of continuity is useful in such scenarios.
+As mentioned, the idea of continuous functions is used in several areas of mathematics, most notably in Topology. A different characterization of continuity is useful in such scenarios.
 ===Theorem===

@@ Line 69: / Line 69: @@
 Readers may note the similarity between this definition to the definition of a limit in that unlike the limit, where the function <math>f</math> can converge to any value, continuity restricts the returning value to be only the expected value when the function <math>f</math> is evaluated. This added restriction provides many new theorems, as some of the more important ones will be shown in the following headings.
-==Operations==
+==Operations/Combinations==
 Since limits are preserved under algebraic operations, let's check whether this is also the case with continuity.

@@ Line 1: / Line 1: @@
 '''Continuity''' marks a new classification of functions, especially prominent when the theorems explained later on in this page will be put to use. However, if one is reading this wikibook linearly, then it will be good to note that the wikibook will describe functions with even more properties than continuity. As an example, the functions in elementary mathematics, such as polynomials, trigonometric functions, and the exponential and logarithmic functions, contain many levels more properties than that of a continuous function. We will also see several examples of discontinuous functions as well, to provide some remarks of common functions that do not fit the bill.
-==Real functions: Continuity at a Point==
+==Continuity at a Point==
 ===Definition===
@@ Line 56: / Line 56: @@
-==Definition==
+==Continuity on an Interval==
 <blockquote style="background: white; border: 1px solid black; padding: 1em;">
 '''Definition of a ''Continuous Function on I'''''<hr/>

← Older revision		Revision as of 17:31, 20 October 2021
Line 1:		Line 1:
	'''Continuity''' marks a new classification of functions, especially prominent when the theorems explained later on in this page will be put to use. However, if one is reading this wikibook linearly, then it will be good to note that the wikibook will describe functions with even more properties than continuity. As an example, the functions in elementary mathematics, such as polynomials, trigonometric functions, and the exponential and logarithmic functions, contain many levels more properties than that of a continuous function. We will also see several examples of discontinuous functions as well, to provide some remarks of common functions that do not fit the bill.		'''Continuity''' marks a new classification of functions, especially prominent when the theorems explained later on in this page will be put to use. However, if one is reading this wikibook linearly, then it will be good to note that the wikibook will describe functions with even more properties than continuity. As an example, the functions in elementary mathematics, such as polynomials, trigonometric functions, and the exponential and logarithmic functions, contain many levels more properties than that of a continuous function. We will also see several examples of discontinuous functions as well, to provide some remarks of common functions that do not fit the bill.
		+
		+	==Real functions: Continuity at a Point==
		+
		+	===Definition===
		+	[[File:Function-1 x.svg\|thumb\|The function <math>f(x)=\tfrac 1 x</math> is continuous on the domain <math>\R\setminus \{0\}</math>, but is not continuous over the domain <math>\R</math> because it is undefined at <math>x=0</math>]]
		+
		+	A [[real function]], that is a [[function (mathematics)\|function]] from [[real number]]s to real numbers, can be represented by a [[graph of a function\|graph]] in the [[Cartesian coordinate system\|Cartesian plane]]; such a function is continuous if, roughly speaking, the graph is a single unbroken [[curve]] whose [[domain of a function\|domain]] is the entire real line. A more mathematically rigorous definition is given below.<ref>{{cite web \| url=http://math.mit.edu/~jspeck/18.01_Fall%202014/Supplementary%20notes/01c.pdf \| title=Continuity and Discontinuity \| last1=Speck \| first1=Jared \| year=2014 \| page=3 \| access-date=2016-09-02 \| website=MIT Math \| quote=Example 5. The function <math>1/x</math> is continuous on <math>(0, \infty)</math> and on <math>(-\infty, 0),</math> i.e., for <math>x > 0</math> and for <math>x < 0,</math> 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 <math>x = 0,</math> and it has an infinite discontinuity there.}}</ref>
		+
		+	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 (mathematics)\|limit]]. First, a function {{math\|''f''}} with variable {{mvar\|x}} is said to be continuous {{em\|at the point}} {{math\|c}} on the real line, if the limit of <math>f(x),</math> as {{mvar\|x}} approaches that point {{math\|c}}, is equal to the value <math>f(c)</math>; and second, the ''function (as a whole)'' is said to be {{em\|continuous}}, if it is continuous at every point. A function is said to be {{em\|discontinuous}} (or to have a ''discontinuity'') at some point when it is not continuous there. These points themselves are also addressed as {{em\|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),</math> with the domain of all real <math>x \neq \pi \frac{2 n + 1}{2},</math> <math>n</math> 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},</math> with the domain of all non-negative reals, has a {{em\|left-hand}} endpoint. In this case only the limit from the {{em\|right}} is required to equal the value of the function. Under this definition ''f'' is continuous at the boundary <math>x = 0</math> 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 {{em\|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
		+	:<math>f : D \to \R \quad</math> be a function defined on a [[subset]] <math> D </math> of the set <math>\R</math> of real numbers.
		+
		+	This subset <math> D </math> is the [[Domain of a function\|domain]] of ''f''. Some possible choices include
		+	:<math>D = \R \quad </math> (<math> D </math> is the whole set of real numbers), or, for ''a'' and ''b'' real numbers,
		+	:<math>D = [a, b] = \{x \in \R \mid a \leq x \leq b \} \quad </math> (<math> D </math> is a [[closed interval]]), or
		+	:<math>D = (a, b) = \{x \in \R \mid a < x < b \} \quad </math> (<math> D </math> is an [[open interval]]).
		+
		+	In case of the domain <math>D</math> being defined as an open interval, <math>a</math> and <math>b</math> do not belong to <math>D</math>, and the values of <math>f(a)</math> and <math>f(b)</math> do not matter for continuity on <math>D</math>.
		+
		+	====Definition in terms of limits of functions====
		+	The function ''f'' is ''continuous at some point'' ''c'' of its domain if the [[limit of a function\|limit]] of <math>f(x),</math> as ''x'' approaches ''c'' through the domain of ''f'', exists and is equal to <math>f(c).</math><ref>{{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</ref> In mathematical notation, this is written as
		+	<math display="block">\lim_{x \to c}{f(x)} = f(c).</math>
		+	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 on the left hand side of that equation has to exist. Third, the value of this limit must equal <math>f(c).</math>
		+
		+	(Here, we have assumed that the domain of ''f'' does not have any [[isolated point]]s.)
		+
		+	====Definition in terms of neighborhoods====
		+	A [[neighborhood (mathematics)\|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 <math>f(c)</math> 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 <math>N_1(f(c))</math> there is a neighborhood <math>N_2(c)</math> in its domain such that <math>f(x) \in N_1(f(c))</math> whenever <math>x\in N_2(c).</math>
		+
		+	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 <math>\exp (1/n)</math> converges to exp(0)]]
		+	One can instead require that for any [[sequence (mathematics)\|sequence]] <math>(x_n)_{n \in \N}</math> of points in the domain which [[convergent sequence\|converges]] to ''c'', the corresponding sequence <math>\left(f(x_n)\right)_{n\in \N}</math> converges to <math>f(c).</math> In mathematical notation, <math>\forall (x_n)_{n \in \N} \subset D:\lim_{n\to\infty} x_n = c \Rightarrow \lim_{n\to\infty} f(x_n) = f(c)\,.</math>
		+
		+	====Weierstrass and Jordan definitions (epsilon–delta) of continuous functions====
		+	[[File:Example of continuous function.svg\|right\|thumb\|Illustration of the <math>\varepsilon-\delta</math>-definition: for <math>\varepsilon = 0.5,c = 2,</math> the value <math>\delta = 0.5</math> 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 <math>f : D \to R</math> as above and an element <math>x_0</math> of the domain ''D'', ''f'' is said to be continuous at the point <math>x_0</math> when the following holds: For any number <math>\varepsilon > 0,</math> however small, there exists some number <math>\delta > 0</math> such that for all ''x'' in the domain of ''f'' with <math>x_0 - \delta < x < x_0 + \delta,</math> the value of <math>f(x)</math> satisfies
		+	<math display="block">f\left(x_0\right) - \varepsilon < f(x) < f(x_0) + \varepsilon.</math>
		+
		+	Alternatively written, continuity of <math>f : D \to R</math> at <math>x_0 \in D</math> means that for every <math>\varepsilon > 0,</math> there exists a <math>\delta > 0</math> such that for all <math>x \in D</math>:
		+	<math display="block">\left\|x - x_0\right\| < \delta \text{ implies } \|f(x) - f(x_0)\| < \varepsilon.</math>
		+
		+	More intuitively, we can say that if we want to get all the <math>f(x)</math> values to stay in some small [[Topological neighbourhood\|neighborhood]] around <math>f\left(x_0\right),</math> we simply need to choose a small enough neighborhood for the ''x'' values around <math>x_0.</math> If we can do that no matter how small the <math>f(x)</math> neighborhood is, then ''f'' is continuous at <math>x_0.</math>
		+
		+	In modern terms, this is generalized by the definition of continuity of a function with respect to a [[basis (topology)\|basis for the topology]], here the [[metric topology]].
		+
		+	Weierstrass had required that the interval <math>x_0 - \delta < x < x_0 + \delta</math> be entirely within the domain ''D'', but Jordan removed that restriction.
		+

	==Definition==		==Definition==

@@ Line 94: / Line 94: @@
 This is the big theorem on continuity.  Essentially it says that continuous functions have no sudden jumps or breaks.
-{{TextBox|'''Theorem'''<hr/>Let f(x) be a continuous function.  If <math>a<b</math> and <math>f(a)<m<f(b)</math>, then <math>\exists c \in (a,b):  f(c) = m</math>.}}
+<blockquote style="background: white; border: 1px solid black; padding: 1em;">
 [[File:Intermediatevaluetheorem.svg|alt=The typical depiction of the intermediate value theorem with one peak and one valley.|thumb|The intermediate value theorem: Given a continuous function on [a,b] and three variables a < c < b it must be the case that ƒ(a) < ƒ(c) < ƒ(b)]]

@@ Line 141: / Line 141: @@
 Assume if possible, <math>M=\sup (f([a,b]))</math> but <math>M\notin f([a,b])</math>.
-Consider the function <math>g(x)=\frac{1}{M-f(x)}</math>. By algebraic properties of continuity, <math>g:[a,b]\to\mathbb{R}</math> is continuous. However, <math>M</math> being a cluster point of <math>f([a,b])</math>, <math>g(x)</math> is unbounded over <math>[a,b</math>, contradicting (i). Hence, <math>M\in f([a,b])</math>. Similarly, we can show that <math>m\in f([a,b])</math>.
+Consider the function <math>g(x)=\frac{1}{M-f(x)}</math>. By algebraic properties of continuity, <math>g:[a,b]\to\mathbb{R}</math> is continuous. However, <math>M</math> being a cluster point of <math>f([a,b])</math>, <math>g(x)</math> is unbounded over <math>[a,b]</math>, contradicting (i). Hence, <math>M\in f([a,b])</math>. Similarly, we can show that <math>m\in f([a,b])</math>.
 ==Appendix==

@@ Line 133: / Line 133: @@
 This is the second part of the theorem. It is the more assertive version of the previous theorem, stating that not only is there a supremum and a infimum, it also is reachable by the function ƒ and will be in between the interval you specified.
-{{TextBox|'''Theorem'''<hr/>Given a continuous function ƒ on [a,b] i.e. <math>f:[a,b]\to\mathbb{R}</math>, if <math>M,m</math> are respectively the upper and lower bounds of <math>f([a,b])</math>, then there exist <math>c,d\in [a,b]</math> such that <math>f(c)=M,f(d)=m</math>.}}
+<blockquote style="background: white; border: 1px solid black; padding: 1em;">
 [[File:Extreme Value Theorem.svg|alt=The typical depiction of continuity: a function with one peak and one valley. f at c and f at d is marked|thumb|A depiction of the Extreme Value Theorem: Given a continuous function on [a,b], there must exist a maximum c and d such that ƒ(c) is the greatest value in the interval and ƒ(d) is the littlest.]]