One-sided derivative ($f_{-}^{\prime }(a),f_{+}^{\prime }(a)$)

The right and left one-sided derivatives are the limits of the difference quotient restricted respectively to positive and negative increments:At an interior point, differentiability is equivalent to existence and equality of these two finite limits. For the absolute value function, $f_{+}^{\prime }(0)=1$ and $f_{-}^{\prime }(0)=-1$, so it is continuous at zero but not differentiable there.