If all derivatives are zero at a point, what does this imply?

Question

Let's say I have a function $f$ which for all positive $n$ and some complex point $z_0$ satisfies $f^{(n)}(z_0) = 0$. What does this say about the function's analyticity or holomorphicity? Obviously, its Taylor series would be identically zero at that point, but does this imply anything about the function in any other region of the complex plane?

Answer 1

The answer to your question depends on if you're considering real or complex functions. The standard real example:

$$ f(x)=\begin{cases}e^{-1/x^2}&x>0\\0&x\leq 0\end{cases}. $$

This real function is infinitely differentiable and the derivative at $0$ is always $0$, however, the function is not identically $0$. This is the standard example of a $C^\infty$ function which is not analytic.

Observe that this situation is not possible for complex differentiable functions as differentiability and analyticity are the same concept. More precisely, any complex differentiable function in a neighborhood of $z_0$ is analytic in a neighborhood of $z_0$, so such a function would equal its Taylor series and be identically zero in a neighborhood of $z_0$.