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User:SamHB/Continuity (view source)
Revision as of 03:45, December 15, 2015
, 03:45, December 15, 2015Restore a sandbox page that had been deleted by an admin in 2009.
{{Template:Math-e}}
Note: This article has wikilinks into my sandbox.
These need to be changed if it is moved to the main article space.
Put simply, a mathematical [[User:SamHB/Function|function]] is '''continuous''' if its graph can be drawn without lifting the pen from the paper. In the figures below, the graph on the left is of a continuous function; the graph on the right is not.
[[Image:Br-cont-function.png]] [[Image:Br-discont-function.png]]
The function on the left is:
:<math>f(x) = x^3 - 3x^2 + 2x + 1\,</math>
The function on the right is:
:<math>f(x) = x^3 - 3x^2 + 2x + 1\,</math> for x <math>\le</math> 2
:<math>f(x) = x^3 - 3x^2 + 2x - 1\,</math> for x <math>>\,</math> 2
==More precise definition==
{{Template:Math-h}}
In the language of [[calculus]], there is a more precise definition in terms of [[User:SamHB/Limit (mathematics)|limit]]s—f(x) is continuous at a point X if the limit of f(x) as x approaches X is equal to the function value at that point, f(X):
:<math>\lim_{x\to X}f(x) = f(X)\,</math>
From the definition of a limit, this means
:For every ε > 0, there is a δ > 0 such that, whenever <math>0 < |x-X| < \delta, |f(x)-f(X)| < \varepsilon\,</math>
In the example in the picture on the right, the function is not contiuous at x=2. We have f(2) = 1, but it isn't continuous there. To see this, let ε=1. We would need to find δ such that
:whenever <math>0 < |x-2| < \delta, |f(x)-1| < 1\,</math>
But for all x with 2 < x < 2.1, we have
:f(x)-1 < -1.769
so
:|f(x)-1| < 1.769
==In topology==
{{Template:Math-a}}
In the language of [[topology]], there is an extremely simple and elegant formulation of continuity. In topology, all questions ultimately refer to open sets. A function is continuous if the inverse image of every open set is open. It is continuous at a point X if the inverse image of every open set containing X is open.
[[Category:Mathematics]]
Note: This article has wikilinks into my sandbox.
These need to be changed if it is moved to the main article space.
Put simply, a mathematical [[User:SamHB/Function|function]] is '''continuous''' if its graph can be drawn without lifting the pen from the paper. In the figures below, the graph on the left is of a continuous function; the graph on the right is not.
[[Image:Br-cont-function.png]] [[Image:Br-discont-function.png]]
The function on the left is:
:<math>f(x) = x^3 - 3x^2 + 2x + 1\,</math>
The function on the right is:
:<math>f(x) = x^3 - 3x^2 + 2x + 1\,</math> for x <math>\le</math> 2
:<math>f(x) = x^3 - 3x^2 + 2x - 1\,</math> for x <math>>\,</math> 2
==More precise definition==
{{Template:Math-h}}
In the language of [[calculus]], there is a more precise definition in terms of [[User:SamHB/Limit (mathematics)|limit]]s—f(x) is continuous at a point X if the limit of f(x) as x approaches X is equal to the function value at that point, f(X):
:<math>\lim_{x\to X}f(x) = f(X)\,</math>
From the definition of a limit, this means
:For every ε > 0, there is a δ > 0 such that, whenever <math>0 < |x-X| < \delta, |f(x)-f(X)| < \varepsilon\,</math>
In the example in the picture on the right, the function is not contiuous at x=2. We have f(2) = 1, but it isn't continuous there. To see this, let ε=1. We would need to find δ such that
:whenever <math>0 < |x-2| < \delta, |f(x)-1| < 1\,</math>
But for all x with 2 < x < 2.1, we have
:f(x)-1 < -1.769
so
:|f(x)-1| < 1.769
==In topology==
{{Template:Math-a}}
In the language of [[topology]], there is an extremely simple and elegant formulation of continuity. In topology, all questions ultimately refer to open sets. A function is continuous if the inverse image of every open set is open. It is continuous at a point X if the inverse image of every open set containing X is open.
[[Category:Mathematics]]