We use the symbols    and    for the left-hand and right-hand limit, respectively, of a function as approaches the point .  The graph of a piecewise continuous function is illustrated in Figure 12.2, where the function is

        

The  left-hand and right-hand limits at ,  ,  and    are easily determined:

    At , the left-hand limit is  ,
and the right-hand limit is .

    At , the left-hand limit is  ,
and the right-hand limit is  .

    At , the left-hand limit is  ,
and the right-hand limit is  . 

        Figure 12.2.  A piecewise continuous function over the interval .

Exploration.

    The  left-hand and right-hand limits at ,  ,  and    are easily determined:

At , the left-hand limit is  ,
and the right-hand limit is .

At , the left-hand limit is  ,
and the right-hand limit is  .

At , the left-hand limit is  ,
and the right-hand limit is  .

(c) 2006 John H. Mathews, Russell W. Howell