= Continuity estimate for the Loewner driving function
{title2=$|\xi_t-\xi_s|\leq C\omega(t-s)+2\omega(2(t-s))$}
Suppose the mapped increments $K_{s,t}$ have real-centred enclosing radii bounded by a nondecreasing modulus $\omega(t-s)\to0$. Nested closures define the unique growing real point $\xi_s$. For $v\in K_{t,t+h}$, its preimage $z=g_{K_{s,t}}^{-1}(v)$ belongs to $K_{s,t+h}$. The uniform displacement estimate for a <mapping-out function> gives $|v-z|\leq C\omega(t-s)$, while $|z-\xi_s|\leq2\omega(t-s+h)$. Letting $h\downarrow0$ with $h\leq t-s$ proves the displayed estimate. It proves uniform <continuity> of the <Loewner transform> on each bounded time interval without evaluating a conformal map on a rough boundary.
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