Taylor series of a function may converge to another function that agrees with it only at the center

Statement

It is possible to have an everywhere infinitely differentiable function $f$ and a point $x_{0}$ in the domain such that the sum of the Taylor series of $f$ at $x_{0}$ exists everywhere but is not equal to $f$ on any interval of positive radius centered at $x_{0}$. In fact, we can arrange our example so that the power series sum agrees with $f$ only at $x_{0}$. Note that if this happens, then there cannot be any other power series centered at $x_{0}$ that converges to $f$ on a positive radius of convergence.

Proof

Consider the function:

$f(x):=\left\lbrace {\begin{array}{rr}\exp(-1/x^{2}),&x\neq 0\\0,&x=0\\\end{array}}\right.$

and the point $x_{0}=0$.

We note that:

• For all $k$, the $k^{th}$ derivative $f^{(k)}(0)$ is 0.
• The Taylor series of $f$ at $0$ is the zero Taylor series, i.e., all the coefficients are zero.
• The Taylor series of $f$ at $0$ converges everywhere to the zero function.
• The only point $x\in \mathbb {R}$ for whuch the Taylor series of $f$ at $0$, evaluated at $x$, converges to $f(x)$, is $x=0$.