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?