I am reading Tu's book "An Introduction to Manifolds" and I have a question related to Taylor's theorem with remainder (which is not related to this post: Taylor's theorem with remainder). So, apparently the theorem states that there exist functions $g_1(x),...,g_n(x)\in C^{\infty}(U)$, such that $$f(x)=f(p)+\sum_{i=1}^n(x^i-p^i)g_i(x),\ \ g_i(p)=\frac{\partial f}{\partial x^i}(p)$$ for a $C^{\infty}$ function $f$ on an open subset $U$ of $\mathbb{R}^n$ star shaped with respect to a point $p=(p^1,...,p^n)$.
From this statement, I understand that any function that is $C^{\infty}$ on any star shaped open set of $\mathbb{R^{n}}$ can be approximated by the value of the function at the point $p$ plus first order corrections that are in agreement with the Taylor expnansion (or at least with the first term of the Taylor expansion) of the same function $f$ if it were analytic in the star shaped open set. So, no corrections that are of order $\mathcal{O}[(x^i-p^i)(x^j-p^j)]$ (i.e. second order) or beyond.
However, right after stating the Taylor's theorem with remainder and right after proving it, the author proceeds in claiming that when the dimensionality is $n=1$ and the point is $p=0$, the function $f$ admits corrections that are beyond the first order, i.e. $$f(x)=f(0)+g_1(0)x+g_2(0)x^2+...+g_i(0)x^i+g_{i+1}(x)x^{i+1}$$ So, f(x) can be expanded as a polynomial with terms that are in agreement with Taylor's series. What am I missing here? Isn't this the same as saying that the function $f$ is real-analytic? If not, how is it different?
Any help will be appreciated. Thanks!