[see also: examination, inspection, verification]
It is an elementary check that $A$ is a vector space.
[see also: examine, verify, test]
It is easily checked that $f$ is l.s.c.
The continuity of $f$ is readily checked, and we can then apply Morera's theorem.
The only thing to check is that maps compose correctly; but this is an easy, if tedious, verification.
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