I'm glad you asked because, as it turns out, continue is not even needed in this case. Here is the simplified code:

while (<>) {
    print "$.\n";
    close ARGV if eof;
}
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I never use continue either; I just copied the code snippet (and tested it!) from the eof documentation. But, that code also used a next statement, which I deleted since it was not needed for your purposes. I should have also deleted the continue statement. I Read The Free Manual, as I'm sure you also did, which explains the relationship between continue and next.