= Oscillating annuli obstruct differentiation at a point
Let $a_n=2^{-2^n}$ and, in angular coordinates near zero, take a bounded indicator alternating between one and zero on the rings $a_{n+1}<|t|\leq a_n$. Since $a_{n+1}/a_n\to0$, the mean over $[-a_n,a_n]$ approaches the value of the outermost ring. Its even and odd subsequences approach one and zero. Thus an integrable function can fail to have a limiting centered average at a specified point, although <Lebesgue differentiation theorem> ensures convergence almost everywhere.
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