Can a function differ from its Taylor series but converge like its Taylor series?

Question

Given a sequence $(a_n)$ can there be two functions $f$ and $g$ satisfying for every $n\in \mathbb N$ the conditions :

$f(x)=a_0+a_1x+\dots + a_nx^n + o_0(x^n)$

and

$g(x)=a_0+a_1x+\dots + a_nx^n + o_0(x^n)$

and nonetheless

$\forall x\neq 0 \quad f(x)\neq g(x)$

?

Answer 1

Yes, the classic example is $f(x)=\begin{cases} e^{-1/x^2} & x \neq 0 \\ 0 & x=0\end{cases}$ and $g(x)=0$.