Art of Problem Solving
AoPS Online
Math texts, online classes, and more
for students in grades 5-12.
Visit AoPS Online ‚
Books for Grades 5-12 Online Courses
Beast Academy
Engaging math books and online learning
for students ages 6-13.
Visit Beast Academy ‚
Books for Ages 6-13 Beast Academy Online
AoPS Academy
Small live classes for advanced math
and language arts learners in grades 2-12.
Visit AoPS Academy ‚
Find a Physical Campus Visit the Virtual Campus

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...)
 
m (This should be merged with continuous)
Line 15: Line 15:
  
 
{{stub}}
 
{{stub}}
 +
{{wikify}}
  
 
[[Category:Calculus]]
 
[[Category:Calculus]]

Revision as of 15:05, 17 July 2008

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 $\mathbb{R}$

Definition

Let $A\subset\mathbb{R}$

Let $f:A\rightarrow\mathbb{R}$

Let $c\in A$

We say that $f$ is continous at point $c$ iff $\forall\epsilon>0\;\exists\delta>0$ such that $x\in V_{\delta}(c)\implies f(x)\in V_{\epsilon}(f(c))$

If $f$ is continous at $c$ $\forall c\in A$, we say that $f$ is Continous over $A$

This article is a stub. Help us out by expanding it. Template:Wikify

Retrieved from "https://artofproblemsolving.com/wiki/index.php?title=Continuity&oldid=27066"