This question actually arises from this answer to another question, which contains the sentence
A function is smooth is it has derivatives of infinite order.
While the author surely didn't actually mean $\frac{\mathrm d^\infty f}{\mathrm dx^\infty}$, a comment on that post got me to think about that.
I've come to the following definition:
Be $\alpha$ an arbitrary ordinal. Then the $\alpha$th derivative of $f(x)$ is defined as follows, provided the used derivatives/limits exist: $$\frac{\mathrm d^\alpha f}{\mathrm dx^\alpha} = \begin{cases} f & \text{if }\alpha = 0\\ \frac{d}{dx}\,\frac{\mathrm d^\beta f}{\mathrm dx^\beta} & \text{if } \alpha=\beta+1\\ \lim_{\beta\to\alpha}\frac{\mathrm d^\beta f}{\mathrm dx^\beta} & \text{if } \alpha \text{ is a limit ordinal} \end{cases}$$ Here the limit is meant pointwise.
Here are a few cases:
If $f(x)$ is an arbitrary polynomial, $\frac{\mathrm d^\omega f}{\mathrm dx^\omega}=0$.
If $f(x) = \exp(a x)$, then $$\frac{\mathrm d^\omega f}{\mathrm dx^\omega}= \begin{cases} 0 & \text{if } -1 \le a < 1\\ \exp(x) & \text{if } a = 1\\ \text{does not exist} & \text{otherwise} \end{cases}$$
- $\sin(x)$ and $\cos(x)$ don't have a $\omega$th derivative.
Also the $C^n$ classes can be generalized in a straightforward way: A function is in $\tilde C^\alpha$ if for every $\beta\in\alpha$, the $\beta$th derivative of $f$ exists and is continuous. For finite $n$, we have $C^n=\tilde C^{n+1}$ ($\tilde C^0$ would be all functions). Moreover, $C^\infty = \tilde C^\omega$. If the $\omega$th derivative exists and is continuous, the function is in $\tilde C^{\omega+1}$.
Now my questions:
- Has anyone already considered such derivatives? Are they even useful somewhere?
- Is there an easy way to characterize the functions in $\tilde C^{\omega+1}$?
- Are there functions in $\tilde C^{\omega+1}$ whose $\omega$th derivative is neither $0$ nor $\exp(x)$? (Otherwise the ordinal order derivatives would be rather boring).
