From continuity to differentiability and analyticity- what's next?

Question

Continuity is an intuitive concept. I will not dwell on the precise definitions of continuity and the rest here. Note that differentiability is a more restrictive condition than continuity, while analyticity for complex-valued functions is even more restrictive than differentiability.

To some extent, I understand the motivation behind defining these terms as they are defined right now. My question is: what is the next condition in the sequence

continuity, differentiability(scalar/vector/left-right/partial:all), analyticity $\cdots$?

Does there exist a next term in this sequence? If yes, in what context? if no, what is the reason? are all possible restrictions on functions' behavior covered in some sense? As one goes on in higher dimensions, is there some behavior that prompts any further restriction, so to speak?

PS: I am talking in very general terms, with their usual connotations.

Answer 1

No, at least not ones I consider analogous, but you've skipped (infinitely) many steps. All these properties of functions are examples of what is called the differentiability class of a function. For example, a continuous function is of class $C^0$, and a differentiable function is of class $C^1$. More generally, a function is of class $C^k$ if it has continuous $k^{th}$ derivatives. This extends to $C^\infty$, the class of functions with continuous $k^{th}$ derivatives for all $k\geq 0$, i.e. the smooth functions, and analytic functions are said to be of class $C^\omega$.

Answer 2

For complex functions, there is a finer hierarchy for entire functions based on order of growth.

Answer 3

I think Jonas Meyer's comment deserves to be an answer. The progression in the question can be looked at many ways depending on context, but one is as increasing levels of rigidity:

Suppose $f:\mathbb{C}\rightarrow\mathbb{C}$ is a function.

If $f$ is continuous, then $f$'s values in an open set are determined by its values on a dense subset of the set. (But outside of the open set, a lot of different things could be going on.)

If $f$ is differentiable as a function $\mathbb{R}^2\rightarrow\mathbb{R}^2$, then we can deduce more about it from less information about its values. The more times continuously differentiable, the easier it is to deduce things about $f$. (But it is still true that we have nearly no knowledge about what it's doing outside of whatever open set we know something about.)

If $f$ is analytic, then its values in the entire plane are determined by its values on any set that has an accumulation point. For example, its values on any dense subset of any nonempty open set (no matter how small) determine its values everywhere.

In this light, polynomial - linear - constant is a natural continuation of the sequence. Each step represents a class of functions such that the amount of information needed about $f$'s values in order for $f$ to be determined completely, is less than the previous.

If $f$ is polynomial, then its values in the entire plane are determined by its values at any infinite set of points, whether the set has an accumulation point or no. In fact, by a finite set of points, but without further information we don't know how many; but if we have a bound on the degree of the polynomial, for example if we know a polynomial bound on its growth rate, then we can say $f$ is determined by its value at a specific (finite) number of points.

If $f$ is linear, then its values everywhere are determined by its values at 2 points.

If $f$ is constant, evaluation at one point suffices to know everything about $f$.

Answer 4

In a conversation today my advisor mentioned Gevrey class, and I immediately remembered this question. It turns out that functions of this class are always $C^\infty$, but may be non-analytic; conversely, there are $C^\infty$ functions which are not Gevrey. See here: http://en.wikipedia.org/wiki/Gevrey_class