Higher derivatives
Differentiating the derivative map repeatedly: is twice differentiable when is itself differentiable, and inductively times differentiable; when exists and is continuous.
Definition 3.19 (Higher Derivatives)
Let be differentiable on . We say that is twice differentiable if
is differentiable. We similarly define thrice differentiable and times differentiable for inductively.
We say that is -times continuously differentiable, and write , if is times differentiable and
is continuous.
Lower orders are continuous
Differentiability implies continuity of the function itself. Iterating this, if is -times differentiable then is continuous for all : each for is differentiable, hence continuous.
Related
Stated in
- Definition 3.19 (Higher Derivatives)§3.3 Higher Derivatives and Taylor’s Theorem