= Translation continuity in Lp
{title2=$\|f(\cdot+h)-f\|_p\to0\quad(h\to0)$}
For $1\leq p<\infty$ and $f\in L^p(\mathbb R^d)$, translations converge to $f$ in the <Lp space> <norm>. First prove this for continuous compactly supported <functions>, using uniform continuity and a common bounded support. Their density in <Lp space>, together with the norm-preserving translation map and the <triangle inequality>, proves the assertion for arbitrary $f$. It implies convergence of averages over shrinking intervals, a basic <approximate identity> argument. The corresponding assertion fails for general $L^\infty$ functions, such as an <indicator function> with a jump.
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