Infinitely differentiable function
Definition at a point
Suppose is a function defined around a point . We say that is infinitely differentiable at if the following equivalent conditions hold:
- All the higher derivatives exist as finite numbers for all nonnegative integers .
- For every nonnegative integer , there is an open interval containing (possibly dependent on ) such that exists at all points on that open interval containing .
- For every nonnegative integer , there is an open interval containing (possibly dependent on ) such that exists and is continuous at all points on that open interval containing .
Definition on an open interval or union of open intervals
Suppose is a function defined on an interval that is open but possibly infinite in one or both directions (i.e., an interval of the form ). We say that is infinitely differentiable on if is infinitely differentiable at every point of .
Instead of a single open interval, we could consider a union of open intervals, and use the same definition: a function is said to be infinitely differentiable on the union of open intervals if it is infinitely differentiable at every point in that union of open intervals.
The set of infinitely differentiable functions on an interval is denoted .
Definition without point or interval specification
If no point or interval is specified, then saying that is infinitely differentiable means that is infinitely differentiable on its entire domain. This makes sense if the domain is a union of open intervals.