I'm confused how analytic (complex) manifolds can exist. My understanding of a manifold is that you have an Atlas $A$, which is your collection of coordinate charts. Each coordinate chart is a tuple of a neighborhood within your topology and a function that maps that topology to your output space. There exists transition maps between those coordinate charts where they intersect, that allow you to seamlessly jump between charts through their intersections.
However, it seems to me that if you have two real functions (charts that have corresponding domains [neighborhoods]) that are "next to" eachother in your manifold, they have to "overlap" for some interval $(a,b)$, where $a<b$ (i.e. they cannot just overlap at one point). The function of this overlap is your transition map for those given charts.
However, due to the existence of the Identity Theorem, if you want two functions $f$ and $g$ to overlap across some inverval $(a,b)$, then either $f$ or $g$ must be non-analytic. Thus, I don't understand how you could say that all of your charts and transition maps on a manifold are both analytic and smooth.
Clearly these kinds of manifolds can exist, so I'm looking for what is wrong with my understanding.