  When given a class of functions and a finite collection of sets, one might be
interested whether the class in question contains any function whose domain is
a subset of the union of the sets of the given collection and whose
restrictions to all of them belong to this class. The collections with the
formulated property are said to be strongly join permitting for the given class
(the notion of join permitting collection is defined in the same way, but
without the words "a subset of"). Three theorems concerning certain instances
of the problem are proved. A necessary and sufficient condition for being
strongly join permitting is given for the case when, for some $n$, the class
consists of the potentially partial recursive functions of $n$ variables, and
the collection consists of sets of $n$-tuples of natural numbers. The second
theorem gives a sufficient condition for the case when the class consists of
the continuous partial functions between two given topological spaces, and the
collection consists of subsets of the first of them (the condition is also
necessary under a weak assumption on the second one). The third theorem is of a
similar character but, instead of continuity, it concerns computability in the
spirit of the one in effective topological spaces.
