= Differentiation along refining metric partitions
{title2=$\operatorname{avg}_{M_k(x)}|f(y)-f(x)|\longrightarrow0\quad\text{almost everywhere}$}
Assume in addition that the <probability space> is a <metric space>, continuous integrable functions are dense in $L^1$, and the diameters of its partition cells tend to zero at each point. For a continuous $g$, the oscillation average tends to zero because every point in $M_k(x)$ is within its <diameter> of $x$. For $h=f-g$, the upper limit of the oscillation of $f$ is bounded by the partition maximal function of $h$ plus $|h(x)|$. The <first-crossing maximal inequality for refining partitions> and <Chebyshev's inequality> bound the <measure> where this exceeds $2\varepsilon$ by $2\|h\|_1/\varepsilon$. Approximation proves the displayed <almost everywhere> assertion. Refinement supplies the maximal estimate; shrinking <diameter> supplies the approximation argument. Without shrinking <diameter>, a constant partition generally converges only to its cell averages.
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