One-sided limit

The limit of a function at a point from the right or left. Let be a mapping from an ordered set (for example, a set lying in the real line), regarded as a topological space with the topology generated by the order relation, into a topological space , and let . The limit of with respect to any interval is called the limit of on the left, and is denoted by

(it does not depend on the choice of ), and the limit with respect to the interval is called the limit on the right, and is denoted by

(it does not depend on the choice of ). If the point is a limit point both on the left and the right for the domain of definition of the function , then the usual limit

with respect to a deleted neighbourhood of (in this case it is also called a two-sided limit, in contrast to the one-sided limits) exists if and only if both of the left and right one-sided limits exist at and they are equal.

Comments

Instead of (respectively, ) one also finds the notations , (respectively, , ).