continuous almost everywhere versus equal to a continuous function almost everywhere
Although these two statements seem alike, they have quite different meanings. In fact, neither one of these statements implies the other.
Consider the function
This function is not continuous at , but it is continuous at all other . Note that . Thus, is continuous almost everywhere.
Suppose is equal to a continuous function almost everywhere. Let be Lebesgue measurable (http://planetmath.org/LebesgueMeasure) with and such that for all . Since for all and , there exists such that . Similarly, there exists such that . Since is continuous, by the intermediate value theorem, there exists with . Let . Since is continuous, is open. Recall that . Thus, . Since is a nonempty open set, . On the other hand, , yielding that , a contradiction.
Now consider the function
Note that . Thus, almost everywhere. Since is continuous, is equal to a continuous function almost everywhere. On the other hand, is not continuous almost everywhere. Actually, is not continuous at any . Recall that and are both dense in (http://planetmath.org/Dense) . Therefore, for every and for every , there exist and . Since and , it follows that is not continuous at . (Choose any .)
|Title||continuous almost everywhere versus equal to a continuous function almost everywhere|
|Date of creation||2013-03-22 15:58:47|
|Last modified on||2013-03-22 15:58:47|
|Last modified by||Wkbj79 (1863)|