  Consider an integral Brakke flow $(\mu_t)$, $t\in [0,T]$, inside some ball in
Euclidean space. If $\mu_{0}$ has small height, its measure does not deviate
too much from that of a plane and if $\mu_{T}$ is non-empty, then Brakke's
local regularity theorem yields that $(\mu_t)$ is actually smooth and graphical
inside a smaller ball for times $t\in (C,T-C)$ for some constant $C$. Here we
extend this result to times $t\in (C,T)$. The main idea is to prove that a
Brakke flow that is initially locally graphical with small gradient will remain
graphical for some time. Moreover we use the new local regularity theorem to
generalise White's regularity theorem to Brakke flows.
