= Dyadic slope martingale
{title2=$G_n(t)=2^n\bigl(f((k+1)2^{-n})-f(k2^{-n})\bigr)$}
For a real <continuous function> $f$ on $[0,1]$, let $G_n$ be its secant slope on each length-$2^{-n}$ dyadic cell. Regard $([0,1],\mathcal B,dt)$ as a <probability space> and use the <filtration> of half-open dyadic cells with the endpoint $\{1\}$ as a separate null cell. The mean of the two child slopes equals the parent slope, so $(G_n)$ is a <martingale>. Its <integral> gives the <linear interpolation> of $f$ on that grid. If $f$ is <Lipschitz continuous>, these slopes are bounded by its <Lipschitz constant>; the <Lp martingale convergence theorem> then supplies a bounded <integral> density for $f$.
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