Characterization of differentiability
is differentiable at exactly when for some and as ; then .
Let . Then is differentiable at iff and function satisfying as such that
Error term
The condition says for small : near , the function agrees with the affine map up to an error that is small compared with . The function quantifies this error, and the requirement is equivalent to
Whenever is differentiable at , the constant must be .
Proof
Suppose first that such and exist. Then
as , because while is bounded. Hence is differentiable at with .
Conversely, if is differentiable at , choose , so that . Define
Then as and the required equality holds.
This characterization is what makes the chain rule transparent: composing two affine approximations with error terms and produces again an affine approximation with a vanishing relative error.
Related
Stated in
- Lemma 3.6§3.1 Introduction