# Composition in function fields

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 $$\Omega\times \{v\in\Omega\mid\deg(v)>0\}$$?