One-sided continuity.

Definition 5.4.1   The function is left continuous (resp. right) at if it has a left limit (resp. right limit) at and if this limit is equal to .

For example, the integer part function is right continuous but not left continuous at 1 (v.s. figure  fig integer part 2).

Proposition 5.4.2   Let be a function defined over a neighborhood of the real number . The function is continuous at if, and only if, the two conditions hold:
1. the function is left-continuous at ;
2. the function is right-continuous at .

Example 5.4.3   Take . We have:
(i)
If , then . It follows that ; thus is left continuous at 0.
(ii)
If , then . It follows that ; thus is right continuous at 0.
Thus, is continuous at 0; see Figure 4.

Example 5.4.4   Let be given as follows:

Let us compute the one-sided limits of at 0:

As , this shows that is left-continuous at 0 and right-continuous at 0, thus is continuous at 0.

Example 5.4.5   Consider the "integer part" function, i.e. . Let . Then the following holds:

 and

As we know that , we conclude that is right-continuous at but not left-continuous at .

Noah Dana-Picard 2007-12-28