One-sided limit

The limit of a function at a point from the right or left. Let $ f $ be a mapping from an ordered set $ X $( 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 $ Y $, and let $ x _ {0} \in X $. The limit of $ f $ with respect to any interval $ ( a, x _ {0} ) = \{ {x } : {x \in X, a < x < x _ {0} } \} $ is called the limit of $ f $ on the left, and is denoted by

$$ \lim\limits _ {x \rightarrow x _ {0} - 0 } f ( x) $$

(it does not depend on the choice of $ a < x _ {0} $), and the limit with respect to the interval $ ( x _ {0} , b) = \{ {x } : {x \in X, x _ {0} < x < b } \} $ is called the limit on the right, and is denoted by

$$ \lim\limits _ {x \rightarrow x _ {0} + 0 } f ( x) $$

(it does not depend on the choice of $ b > x _ {0} $). If the point $ x _ {0} $ is a limit point both on the left and the right for the domain of definition of the function $ f $, then the usual limit

$$ \lim\limits _ {x \rightarrow x _ {0} } f ( x) $$

with respect to a deleted neighbourhood of $ x _ {0} $( 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 $ x _ {0} $ and they are equal.

Comments

Instead of $ \lim\limits _ {x \rightarrow x _ {0} + 0 } $( respectively, $ \lim\limits _ {x \rightarrow x _ {0} - 0 } $) one also finds the notations $ \lim\limits _ {x \rightarrow x _ {0} + } $, $ \lim\limits _ {x \downarrow x _ {0} } $( respectively, $ \lim\limits _ {x \rightarrow x _ {0} - } $, $ \lim\limits _ {x \uparrow x _ {0} } $).