Let $G_K$ denote the absolute Galois group of a local field $K$ and for $v\in\mathbb{Q}_{\geq -1}$, denote by $G_K^v$ the higher ramifcation groups.
a) Is it true that
$$ G_K^v = \bigcap_{w\leq v} G_K^w \;?$$
b) In particular, I can intuitively see that using continuity properties of the (inverse of the) Hasse-Herbrand function, that we should have
$$G_K^v = \bigcap_{w<v} G_K^w,$$
although in b) it seems that if $v=-1$, then the intersection on the right-hand side is empty since the domain and range of the Hasse-Herbrand function is usually $[-1,+\infty)$. I refer to 0.3.4, equality directly above Prop 0.67 in http://staff.ustc.edu.cn/~yiouyang/galoisrep.pdf
c) Also, is it true that $\cap_{w>v} G_K^w = \{1\}$? In the case that $L/K$ is finite Galois, it is easy to see that the intersection of all higher ramification groups of $\rm{Gal}$ $(L/K)$ is trivial since a finite group has only a finite number of subgroups.