Right and Left Derivatives

We have seen that there is a one-sided limit and one-sided continuity. Here we see one-sided derivatives as well.

Definition

The right derivative of a function at a point x is denoted by , such that

,

provided that this limit exists.

Similarly, the left derivative of a function at a point x is denoted by , such that

,

provided that this limit exists.

The function is said to be right differentiable if the first limit exists, and we say it is left differentiable if the second limit exists.

NB

The function f is differentiable at a point a if both its left and right derivatives exist at that point and are equal. We say that:

Example

Examine the differentiability of the function  at x=0.

Solution

Theorem

If the function differentiable at a point in its domain, it must be continuous at that point.

NB

The converse of this theorem is not true. Remember the absolute value function, which is continuous at 0, however not differentiable there.