Wald Theorem 2.2.1 Problem on p.15

[syhpehtam]

Hi, I have some trouble with Theorem 2.2.1 in Wald's GR book p.15.

He derived formula 2.2.5 by using 2.2.4. Here, the $f$ in $v(f)$ is a map $f:M\rightarrow\mathbb R$, but 2.2.4 is the expression for $f$ only in the domain $O\subseteq M$ and we don't know the expression for $f$ outside $O$. So how can 2.2.5 be valid. $v$ is a map $v:\mathcal F\rightarrow\mathbb R$, but $x^{\mu}\circ\psi$ is a map:$O\rightarrow\mathbb R$ which doesn't belong to$\mathcal F$, so $v(x^{\mu}\circ\psi)$ in 2.2.5 & 2.2.7 doesn't make sense.

Is there anything assumed in advance by the author that make these wrong expressions in the formulas become meaningful?

 

[syhpehtam]

I've found a theorem in another GR book which says:
If $f_1,f_2\in\mathcal F$,and there exists a neighborhood $N$ of $p\in M$ such that $f_1|_N=f_2|_N$, then $v(f_1)=v(f_2)$ for $v\in V_p$.

So my trouble is solved.