Difference between revisions of "Continuity"
(New page: The notion of '''Continuity''' is one of the most important in real analysis, partly because continous functions most closely resemble the behaviour of observables in nature. Although con...) |
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The notion of '''Continuity''' is one of the most important in real analysis, partly because continous functions most closely resemble the behaviour of observables in nature. | The notion of '''Continuity''' is one of the most important in real analysis, partly because continous functions most closely resemble the behaviour of observables in nature. | ||
− | Although continuity and continous functions can be defined on more general sets, we will restrict ourselves to <math>\mathbb{R}</math> | + | Although continuity and continous functions can be defined on more general sets, we will first restrict ourselves to <math>\mathbb{R}</math> |
==Definition== | ==Definition== | ||
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Let <math>c\in A</math> | Let <math>c\in A</math> | ||
− | We say that <math>f</math> is continous at point <math>c</math> | + | We say that <math>f</math> is continous at point <math>c</math> if <math>\forall\varepsilon>0\;\exists\delta>0</math> such that for all <math>x\in A</math>, <cmath>|x-c|<\delta\Rightarrow |f(c)-f(x)|<\varepsilon.</cmath> |
− | If <math>f</math> is continous at <math>c</math> <math> | + | If <math>f</math> is continous at <math>c</math> for all <math>c\in A</math>, we say that <math>f</math> is '''continous over <math>A</math>'''. |
+ | |||
+ | ==Definition for metric spaces== | ||
+ | |||
+ | We can easily extend this definition to [[metric space|metric spaces]]. Let <math>X</math> and <math>Y</math> be metric spaces. Given a function <math>f:X\to Y</math>, and a point <math>c\in X</math>, we say that <math>f</math> is continuous a <math>c</math> if, for all <math>\varepsilon >0</math> there is a <math>\delta>0</math> such that for all <math>x\in X</math>,<cmath>d_X(c,x)<\delta\Rightarrow d_Y(f(c),f(x))<\varepsilon.</cmath> | ||
+ | |||
+ | If <math>f</math> is continous at <math>c</math> for all <math>c\in X</math>, we say that <math>f</math> is '''continous over <math>X</math>''' | ||
+ | |||
+ | ==Definition for Topological spaces== | ||
+ | |||
+ | Perhaps the most general definition of continuity is in the context of [[topological space|topological spaces]]. If <math>X</math> and <math>Y</math> are topological spaces, then a function <math>f:X\to Y</math> is called continuous if for any open set <math>\mathcal{U}</math> in <math>Y</math>, it's ''preimage'' (i.e. the set <math>f^{-1}(\mathcal{U}) = \{x\in X|f(x)\in\mathcal{U}\}</math>) is an open set in <math>X</math>. Note that the image of an open set in <math>X</math> does '''not''' have to be open. | ||
+ | |||
+ | It can be shown that if <math>X</math> and <math>Y</math> are metric spaces under the metric space topology, that this definition of continuity coincides with the previous one. | ||
{{stub}} | {{stub}} | ||
[[Category:Calculus]] | [[Category:Calculus]] | ||
+ | [[Category:Analysis]] | ||
+ | [[Category:Topology]] |
Latest revision as of 15:42, 1 December 2015
The notion of Continuity is one of the most important in real analysis, partly because continous functions most closely resemble the behaviour of observables in nature.
Although continuity and continous functions can be defined on more general sets, we will first restrict ourselves to
Definition
Let
Let
Let
We say that is continous at point
if
such that for all
,
If is continous at
for all
, we say that
is continous over
.
Definition for metric spaces
We can easily extend this definition to metric spaces. Let and
be metric spaces. Given a function
, and a point
, we say that
is continuous a
if, for all
there is a
such that for all
,
If is continous at
for all
, we say that
is continous over
Definition for Topological spaces
Perhaps the most general definition of continuity is in the context of topological spaces. If and
are topological spaces, then a function
is called continuous if for any open set
in
, it's preimage (i.e. the set
) is an open set in
. Note that the image of an open set in
does not have to be open.
It can be shown that if and
are metric spaces under the metric space topology, that this definition of continuity coincides with the previous one.
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