In the SLE paper "Basic Properties of the SLE" from Rohde and Schramm, it is mentioned on page 898 that the map
$$(y,t)\mapsto g_t^{-1}(iy+\xi(t))$$
is clearly continuous on $y>0,t>0$, where $g_t$ are the Loewner flow, and $\xi(t)$ is a Brownian motion. Do i miss an analytic fact since this is clear? I know that the map is differentiable for fixed z and conformal for fixed t, but theres still some way to go for proving nice properties for the two-variable map.