In Renteln's, Manifolds, Tensors and Forms, p. 81, The cotangent space as a jet space$^*$, we have the following definitions
Let $f:M \to \mathbb R$ be a smooth function, $p \in M$, and $\{x^i\}$ local coordinates around $p$. We say the $f$ vanishes to first order at $p$ if $\partial f/\partial x^i$ vanishes at $p$ for all $i$.
But there is no requirement on $f(p)$ to vanish. Moreover
Inductively, for $k \ge 1$ we say that $f$ vanishes to $k$th order at $p$ if, for every $i$, $\partial f/\partial x^i$ vanishes to $(k-1)$th order at $p$, where $f$ vanishes to zeroth order at $p$ if $f(p) = 0$. Put another way, $f$ vanishes to $k$th order at $p$ if the first $k$ terms in its Taylor expansion vanish at $p$.
I'm not sure whether he meant "orders" instead of "terms" in the last sentence, or whether the "first $k$" counts from 0 or from 1, but he continues
For $k > 0$ let $M_p^k$ denote the set of all smooth functions on $M$ vanishing to $(k − 1)$th order at $p$, and set $M_p^0 := \Omega^0(M)$ and $M_p := M_p^1$. Each $M_p^k$ is a vector space under the usual pointwise operations, and we have the series of inclusions
$$
M_p^0 \supset M_p^1 \supset M_p^2 \dots
$$
So $M_p = M_p^1$ is the set of functions vanishing to zeroth order at $p$, i.e. all those with $f(p) = 0$. $M_p^2$ is the set of those with $\partial f/\partial x^i = 0$ but not necessarily $f(p) = 0$. So the inclusion are not correct? Finally
We now define $T_p^*M$, the cotangent space to $M$ at $p$, to be the quotient space
$$
T_p^*M = M_p/M_p^2.
$$
Which is the set of equivalence classes, each of which consists of functions different from each other by a function vanishing to the 1st-order (with zeroth 1st partial derivative).
Question: Is it needed or not that $f \in M_p^k, k \ge 1$ should have $f(p) = 0$ ? If not then I suppose it's possible to define the cotangent space simply as
$$
T_p^*M = \Omega^0(M)/M_p^2.
$$
Best Answer
Physor is correct. The condition that $f(p)=0$ was inadvertently omitted from the definition of vanishing to first order. (I have updated the list of errata to include this.)
Also, Yuri is correct that the phrase `` …the first $k$ terms in the Taylor expansion…’’ means to start counting at $k=0$ for the constant term, $k=1$ for the linear term, etc..
I agree that the discussion in that section probably ought to have been rewritten to make it clearer. Perhaps if there’s ever a second edition…