Let $k=\mathbb F_q\left(\!\left(\frac1T\right)\!\right)$. One has the map: $\circ:k\times\{v\in k\mid\deg(v)>0\}\to k$ defined by $f\circ g=\sum_{n\ge-m}a_ng^{-n}$ where $f=\sum_{n\ge-m}a_n\frac1{T^n}$ ($m\in\mathbb N_0$, $a_n\in\mathbb F_q$). Denote by $\Omega$ a completion of $\overline k$ for the valuation $-\deg$. Can one extend $\circ$ continuously to $\Omega\times \{v\in\Omega\mid\deg(v)>0\}$?
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