I am working on a proof and I would like to make the following assertion:
If a real function $f: [z,\infty) \to (z,\infty)$ is analytic on $[z,\infty)$, and I know that $f(z) \neq 0$, then $f$ is analytic at $z$.
(I know that f is completely monotone on its domain.) Any help with thinking about whether or not this assertion may be true would be greatly appreciated. Thank you!
So I think the answer to my question is "no." For $f$ to be analytic at $z$, $z$ would have to be an interior point of $[z,\infty)$, which is not. It would be great if I were wrong. Thanks!
An analytic function $f$ on an open interval can be continuously extended to the boundary if $f$ is bounded. However this doesnt imply that $f$ would be analytic in the boundary.
A classical example is the function
$$f:(0,\infty)\to\Bbb R,\quad x\mapsto e^{-1/x}$$
continuously extended to the point zero with $f(0):=0$. Then the extended $f$ is smooth but it is not analytic at zero because $f^{(k)}(0)=0$ for all $k\in\Bbb N$.
Im not sure if this answer your question.