Corollary 2

Suppose G compact. Then G is totally discontinuous if and only if $\widehat{G}$ is a torsion group.

The group G is totally discontinuous if and only if its connected component is reduced to $\{e\}$; Prop. 4 shows that this condition is equivalent to $B_{\widehat{G}}=\widehat{G}$. Since the group $\widehat{G}$ is discrete (Prop. 18 of II, p. 233), this amounts to saying that every element of $\widehat{G}$ generates a finite group, hence that $\widehat{G}$ is torsion.