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Sorry, but I do not see how the final assertion on the measure of $A$ in $B_r(x_0)$ follows for every $r<r_n$, with your construction (it follows immediately only for SOME $r$). Moreover, the intersection of the sets $A_k$, for $k=1,...,n$ is simply $A_n$ (considering simply $A_k$ the set where $|f(x)−f(x_0)|<1/2^k$, or am I wrong?).