Encyclopedia > Continuity property

  Article Content

Continuity property

THEOREM: Suppose that f : [a , b] -> R is continuous on the closed bounded interval [a , b]. Then the image f([a , b]) is also a closed bounded interval.

This theorem contains three assertions:
(i) The image set is an interval. This is the Intermediate Value Theorem.
(ii) This interval is bounded, so that f is bounded.
(iii) This bounded interval is closed, so that f attains both its bounds.



All Wikipedia text is available under the terms of the GNU Free Documentation License

 
  Search Encyclopedia

Search over one million articles, find something about almost anything!
 
 
  
  Featured Article
Bullying

... a legitimate basis of authority. The first to have the title of "Tyrant" was Pisistratus in 560 BC. In modern times Tyrant has come to mean a dictator who rules ...

 
 
 
This page was created in 23.3 ms