We do not exclude the possibility that A consists of precisely the polynomials.
The resulting metric space consists precisely of the Lebesgue integrable functions, provided we identify any two that are equal almost everywhere.
Precisely r of the intervals Ai are closed.
Thus A and B are at distance precisely d.
We have d(f,g)=0 precisely when f=g a.e.
Important analytic differences appear when one writes down precisely what is meant by......
More precisely, f is just separately continuous.
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