01.06.2012

Let be infinitely differentiable functions, where . In this post I will show some closure properties of this class of functions, such that it becomes easy to see whether a given composite function is infinitely differentiable (but not necessarily that it isn’t). This is of interest when studying the theory of generalized functions.

The derivative of is infinitely differentiable:

By induction, this also shows that any finite derivative of is infinitely differentiable.

The product is infinitely differentiable. By the product rule,

Applying this rule again,

A pattern emerges. By induction one shows the general Leibniz rule:

for .

By induction, this also shows that , where , is infinitely differentiable (the cases and are trivial).

The linear combination , where , is infinitely differentiable. By linearity of differentiation,

By induction one shows

for .

By induction, this also shows that any finite linear combination is infinitely differentiable.

The composition , where the range of is contained in the domain of , is infinitely differentiable. By writing down the first few formulas, observing a pattern (not easy), and again using induction, one shows the FaĆ di Bruno’s formula for the :th derivative of .

By induction, this also shows that any finite composition of infinitely differentiable functions is infinitely differentiable.

The inverse is infinitely differentiable, where . Assume is -times differentiable, with . Using the general Leibniz rule,

It follows that if is -times differentiable, then the :th derivative must be given by

Since the :th derivative is given in terms of lower derivatives, this formula can be proved to hold by induction.

In particular, this shows that is infinitely differentiable for .