Is the composition of class $C^k$ functions also of class $C^k$?
I feel like this one should be true but I can't find any reference.
Is the composition of class $C^k$ functions also of class $C^k$?
I feel like this one should be true but I can't find any reference.
Yes, this is a consequence of the chain rule, which allows us to calculate $(f\circ g)^{(k)}$ in terms of $f,f',\ldots,f^{(k)},g,g',\ldots,g^{(k)}$.
Yes this is true.. If $f$ and $g$ are $C^1$ function, we have $$d(f\circ g)_x= d f_{g(x)}\circ dg_x$$
so we have $f\circ g$ is also $C^1$, Now take $f'$ and $g'$ which is also $C^1$, you will have for $h=f'\circ g'$, $h'$ is also $C^1$... proceed inductively... you will get.