One-sided derivative

A generalization of the concept of a derivative, in which the ordinary limit is replaced by a one-sided limit. If the following limit exists for a function $ f $ of a real variable $ x $:

$$ \lim\limits _ {x \rightarrow x _ {0} + 0 } \ \frac{f ( x) - f ( x _ {0} ) }{x - x _ {0} } \ \ \left ( \textrm{ or } \ \lim\limits _ {x \rightarrow x _ {0} - 0 } \ \frac{f ( x) - f ( x _ {0} ) }{x - x _ {0} } \right ) , $$

then it is called the right (respectively, left) derivative of $ f $ at the point $ x _ {0} $. If the one-sided derivatives are equal, then the function has an ordinary derivative at $ x _ {0} $. See also Differential calculus.