For the real function f, if

exists, this limit is the left derivative of f at a. Similarly, if

exists, this limit is the right derivative of f at a. (See limit from the left and right.) The derivative f′(a) exists if and only if the left derivative and the right derivative of f at a exist and are equal. An example where the left and right derivatives both exist but are not equal is provided by the function f, where f(x) = |x| for all x. At 0, the left derivative equals−1 and the right derivative equals+1.