One-sided limit at one point.

Definition 4.2.1
1. We say that the function has a right limit at and we denote if

2. We say that the function has a left limit at and we denote if

We define the integer part function'':

Definition 4.2.2   For any real number , there exists a unique integer such that . The integer is called the integer part of and is denote by (v.i. Figure 6).

Let now . Then: and .

Therefore the integer part function has no limit at 0, by Proposition 1.2. Similar definitions can be written for infinite one-sided limits. Please try to do it.

For example, if , then and .

Proposition 4.2.3   The function has a limit at if, and only if, the three following conditions are fulfilled:
• has a left limit at ;
• has a right limit at ;
• .

For example, let . Then and . Hence, has no limit at 0 (see Figure  7).

Noah Dana-Picard 2007-12-28