generalized Darboux function
Darboux proved (see parent object) that if is differentiable then is a Darboux function. The class of Darboux functions is very wide. It can be shown that any function can be written as a sum of two Darboux functions. We wish to give more general definiton of Darboux function.
It can be easily proved that connected subsets of intervals (in ) are exactly intervals. Thus this definition coincides with classical definiton, when is an interval and .
Note that every continuous map is a Darboux function.
Also the composition of Darboux functions is again a Darboux function and thus the class of all topological spaces, together with Darboux functions forms a category. The category of topological spaces and continuous maps is its subcategory.
|Title||generalized Darboux function|
|Date of creation||2013-03-22 19:18:36|
|Last modified on||2013-03-22 19:18:36|
|Last modified by||joking (16130)|