Let $G$ be a locally compact group. Denote by $C(G)$ and $C_{0}(G)$ respectively the spaces of continuous bounded functions on $G$ and the continuous bounded functions on $G$ which vanish at infinity.
Let $(f_{\alpha})$ be a net in $C_{0}(G)\subset L^{\infty}(G)$.
If $f_{\alpha}$ converges in the weak-$*$ topology to $f$, need $f$ be in $C_{0}(G)$? or need it even be continuous at all?
The limit need not be continuous: If $f_n \in C_c ((0,1)) $ with $0\leq f_n \leq 1$ and with $f_n \equiv 1$ on $(1/n, 1-1/n) $, then $f_n \to 1_{(0,1)}$ almost everywhere and thus weak-$\ast $ in $L^\infty (\Bbb {R}) $ (by the dominated convergence theorem).
Also, the limit need not vanish at infinity, as one can see by choosing $f_n \equiv 1$ on $(-n,n) $ and $f_n \in C_c (\Bbb {R}) $ and $0\leq f_n \leq 1$.
Even more is true for countably infinite discrete groups. In that case $L_\infty(G)$ is the second dual of $C_0(G)$. By Goldstine's theorem (and metrisability of the weak*-topology on bounded sets), every element of $L_\infty(G)$ is a limit of a weak*-convergent sequence from $C_0(G)$.