Question about Taylor's series

Question

Is there an example of a function whose taylor series converge at every point but does not equal the value of the function at every point?

Answer 1

The standard example is $f(x)=e^{-1/x^2}$ when $x\ne 0$, and $f(0)=0$. The Maclaurin series of this is identically $0$. So there is convergence to $f(x)$ only at $x=0$.

We cannot do better (or should one say worse?): The Taylor series at $x=a$, if it exists, converges to $f(x)$ at $x=a$. So there is always convergence to $f(x)$ at at least one point.

Answer 2

One well-known example is $$ f(x) = e^{-\frac{1}{x^2}} \text{.} $$ This $f$ is differentiable arbitrarily often, and $f^{(n)}(0) = 0$ for all $n$, meaning its taylor series around $0$ is identically zero. Yet $f(x) \neq 0$ for all $x \in \mathbb{R} \setminus \{0\}$.