The *x*-axis points of Heath's *V*-space at which *f* is continuous
is of course the join of all the *V*_{i, n} — a countable
join of countable sets — hence itself countable, while that *x*-axis
is not. So the section *f* cannot be continuous, i.e.,
no section for *p* can be continuous.

Of interest to topos theorists ...
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