$\DeclareMathOperator\dom{dom}$Sorry to bother the community again with these type of questions about power series, I am ready to delete the question if it is not suitable.


**Definition:** I say a function $h$ is $(m,n)$-representable by power series iff $0\in \dom(h)$, $h(0)=0$, and $h$ is a restriction (to a smaller domain) of some power series (centered at $0$) that takes as input $m$ real variables and outputs a vector in $\mathbb{R}^n$. In other words, $0\in \dom(h)\subseteq B\_r(0)$ (where $r$ is the radius of convergence of the power series), and for every $x\in \dom(h)$ we have that $h(x)$ equals the evaluation of the power series at the point $x$.


**Question:** Let $f$ be $(a,b)$-representable by power series. Let $g$ be $(b,c)$-representable by power series. It is given that $\dom(g\circ f)$ (i.e. $\{x\in \dom(f)\mid f(x)\in \dom(g)\}$) is path connected. It is also given that $\dom(f),\dom(g)$ are path connected. Must $g\circ f$ be $(a,c)$ representable by power series?


I edited my question to include the hypothesis of path connectivity of domains of $f,g$



Composition of power series is power series?
Maybe I misunderstood the question, is this a counterexample?


Let $c>1$, put $f(x) = \sin x$ and $g(x) = \tfrac{x}{c^2+x^2}$. Then $f$ is representable by a power series on $\mathbb{R}$, $g$ is representable by a power series on $(-c,c)$ and $f(\mathbb{R}) \subset (-c,c)$.


However, the meromorphic function $g \circ f$ has a pole at $i \sinh^{-1}(c)$, so the power series of $g \circ f$ is only convergent on $(- \sinh^{-1}(c), \sinh^{-1}(c))$, not on all of $\mathbb{R}$.


