I'm reading textbooks on manifolds. I usually see a saying that "function $f$ is $C^k$ at a point $p \in M$". I am confused with this. What does this mean?
Furthermore if $f$ is $C^k$ at $p$, is it necessary that $f$ is $C^k$ in some neighborhood of $p$?
Thanks for your help in advance.
On
It means the $k$th differential of the function $f$ is continuous at $p$. This means the the function has to be differentiable around p but does not mean it is differentiable k times everywhere on the manifold (see https://en.wikipedia.org/wiki/Differentiable_function)
By definition, a function $f: M \rightarrow \mathbb{R}$ is $C^k$ if there exists a chart $(U, \text{x})$ around $p$ such that $f \circ \text{x}^{-1}$ is a $C^k$ function on the open set $\text{x}(U) \subset \mathbb{R}^n$. So if a function is $C^k$ at a point, it is necessarily true that the function is $C^k$ in some neighborhood of the point: that is because what you're actually doing is writing the function in local coordinates of the manifold in order to apply the definition of differentiability that is known for euclidean spaces, and that definition imply taking the difference between two values of a function in arbitrarily close points. That is why afterall, that for euclidean spaces, differentiable function are definied mostlty on open sets.