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Real-valued functions of real variable, analytic properties of functions and sequences, limits, continuity, smoothness of these.
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A lower estimate of the derivative of a distance function
Let $K$ be any closed set in $\mathbb{R}^d$. In your case, $K= \partial D \cap B(x,r)$ but it doesn't matter. Let $F = d(\cdot, K)$.
Let $x$ be a point of differentiability of $F$ not in $K$. Let $y …