# Differentiability on an interval.

Definition 6.4.1   The function is differentiable on the open interval if it is differentiable at each point of .

Definition 6.4.2   The function is differentiable on the closed interval if it verifies the following conditions:
1. is differentiable on the open interval ;
2. is differentiable on the right at ;
3. is differentiable on the left at .

There are similar definitions for other intervals, like , etc.

Example 6.4.3   Take .

For , .

Hence, the square root function is differentiable on .

Take now . Then:

The square root function is not differentiable on the right at 0.

Theorem 6.4.4 (Table of usual derivatives)

 conditions 0 1 , when x

Noah Dana-Picard 2007-12-28