Whether differentiable means smooth in do Carmo's Riemannian Geometry

Question

Picture below is from the 26th page of Do Carmo's Riemannian Geometry. In my view, by (5), if $f\in C^k$, then $Xf\in C^{k-1}$. Therefore, the last red line should be wrong. But in this book, there is not $C^k$. I guess the differentiable function is smooth function (namely, $f\in C^\infty$), is it ?

Answer 1

The index is your friend! A quick look at the index turns up the entry "Differentiable structure" pointing to page 2, where you'll find the following sentence:

Differentiable always signifies of class $C^\infty$.