Thread: Differentiability classes

Take the polynomial . Is of differentiability class or ?

There's more to this, but I hope to figure it out with an answer to this example...

All polynomials are in , since the k-th derivative of any polynomial is defined and continuous for all k.

So then you do just consider 0 as the k-th derivative of the polynomial, and consider that 0 is infinitely and continuously differentiable? I just wanted to question the case of when it hits zero, as with polynomials...

yeah, I suppose the derivative of 0 is still zero. makes sense if you think of it like a chart of (for motion) position versus velocity versus acceleration. if you aren't moving, then the velocity graph is zero, and since your velocity isn't changing, your acceleration is zero. I would figure it wouldn't matter how far out you go, if you have any countinuous, differentiable curve, then I would imagine you could always take k-th derivative, as no matter how many times you take it, you should still get a nice smooth curve with no ugly discontinuous irregularities (sorry about the spelling)

Originally Posted by Chemboy

So then you do just consider 0 as the k-th derivative of the polynomial, and consider that 0 is infinitely and continuously differentiable? I just wanted to question the case of when it hits zero, as with polynomials...

Yes. 0 is just a constant function and it is differentiable.

Polynomials are more than just infinitely differentiable.

Beyond there is a class of functionns called "analytic". Analytic functions are functions that are locally representable as a convergent power series. A power series is a generalization of a polynomial (sort of a polynomial of possibley infinite degree) and analytic functions are generalizations of polynomial functions. In some sense they are "nicer" than just infinitely differentiable functions. Polynomials are particularly simple analytic functions.

The subject of analysis with one complex variable is basically the study of complex analytic functions. They have some surprising properties. Real analytic functions share some, but not all of those properties.

If you are studying objects like manifolds you are likely to also see some mention made of analytic manifolds.