Is it meaningful to say a function is of class $C^n$ at a point? I am asking because I've seen in the books they say a function is of class $C^n$ on a set (not at a point).
 A: It is "meaningful" in the sense that it makes sense. It simply means a function that is $n$ times continuously differentiable at that point. But is it useful?
Obviously it makes sense to talk about continuity at a single point. Consider something like$$f(x) = \begin{cases}x, x\ \mathrm{is\ irrational}\\ -x, x\ \mathrm{is\ rational}\end{cases}$$Clearly $f$ is continuous at $0$ but nowhere else.
Similarly, one can construct a function which is differentiable at exactly one point $x$ but nowhere else. Such a function must also be continuous at $x$, i.e. $f(x+h)$ as $h\rightarrow 0$ must be equal to $f(x)$.
How about a function $f$ which is twice differentiable at exactly one point $x$? Suppose $f'$ is defined only at $x$. Then it is meaningless to speak of limits of $f'(x+h)$, and the second derivative cannot be defined. Therefore, by contradiction, there must be some neighbourhood around $x$ for which the first derivative is defined.
This intuitively establishes that if a function is $n$ times differentiable at a point, it is $n-1$ times differentiable in some neighbourhood around it.
How about continuity of this last derivative? Because it's only defined at a single point, it ends up being "technically continuous".
This gives us a useful result: If a function is $C^n$ at a point for some $n>1$, it must be $C^{n-1}$ on some neighbourhood of that point. This suggests the concept of "$C^n$ at a point" isn't useful, because $C^{n-1}$ on a neighbourhood is a more useful fact to work with (usually).
