Does there exist a function $f(z)$ which is analytic only for $\Im(z)\ge 10$ and nowhere else ?
Intuitively I guess that there does not exist such function ! But there is a function which is analytic only when $\Im(z)>10$ and nowhere else. For example we can define $$f(z)=\begin{cases}e^z &\text{ , when $\Im(z)>10$ }\\ \overline z &\text{ , elsewhere}\end{cases}$$ which is analytic only when $\Im(z)>10$ and nowhere else.
But for the question $\Im(z) \ge 10$ how can I prove there does not exist such function ?
My understanding of the question is as follows: can we have a function analytic for $\Im z >10$ such that for each $z$ with $\Im z=10$ there is an open disk around $z$ on which $f$ is analytic but $f$ is not analytic in any disk on which $\Im z <10$. [Analytic at a point $z$ with $\Im z=10$ means analytic in a neighborhood of that point]. This is clearly impossible: there is a neighborhood of $10i$ in which $f$ is analytic and this neighborhood contains an open disk contained in $\{z: \Im z <10\}$.