So I'm teaching myself from a MultiVariable Calculus text and there was an extremely brief definition of the class of a function. I understand the basic concepts of the notation but had a few questions that weren't addressed by the books one paragraph definition. They are as follows:

(1) Does the notation imply: $\displaystyle C^{n} \subset C^{n+1} \subset C^{\infty}$? So that a function of class infinity could be represented as class n?

(2) While a polynomial function, $\displaystyle f($x$\displaystyle )$ is of class $\displaystyle C^{n}$, such that $\displaystyle D^{n+1}f = 0$, is it's class capped at n? or is it considered of class infinity since a derivation of zero exists and is zero?

2. For (1), you have written it backward, but otherwise your understanding is correct:

$\displaystyle C^\infty \subset C^{n+1} \subset C^n$

For (2), your understanding is also correct. The function $\displaystyle f$ would be a member of class $\displaystyle C^\infty$. In fact, every polynomial function has the property of being a smooth function, which is another way of saying that it belongs to class $\displaystyle C^\infty$.