Take $K$ to be a finite (complete) extension of $\mathbb{Q}_p$. Consider two power series $f(x), g(x) \in K[[x]]$ such that $g(0)$ has positive valuation. For this question let's assume $h(x) = f(g(x))$ is defined as a formal power series, and has some radius of convergence.
Now suppose the radius of convergence of $f$ is $r_2 > 0$ and the radius of convergence of $g$ is $r_1>0$. Suppose also $x_0$ is a point such that $g(x_0)$ converges and $|g(x_0)| < r_2$. Does this imply $h(x_0)$ must also converge, or do there exists examples of series $f,g$ satisfying the above where $h(x_0)$ does not converge?
Just for some context if I understand correctly we may sometimes have the function $h$ is defined at a point $x_0$, yet it is not equal to $f$ applied to $g(x_0)$. I am just trying to get a better idea of how poorly behaved composition of p-adic power series might be.
Edit: thanks to reuns comment I see we need $f$ to converge on some neighborhood of $g(x_0)$. If we assume this condition is true is it enough to get that $h(x_0)$ converges?
Edit 2: maybe it is better to ask if this question is true when $g(0) = 0$, and the composition of formal power series is actually defined. So let's ask the same question now assuming $g(0) =0$.