Hi, please see How do I post a question effectively? -- it is much easier to help if we can see your code.

In the subroutines should I connect and disconnect the database every 
+time
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No, you should not connect to the DB every time you call a sub, if you can help it: DB connections can be expensive.

can connection gone ?
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You can open your DB connection and store the handle in a package variable:

#!/usr/bin/perl
use DBI;
my $dbh = DBI->connect("dbi:SQLite:dbname=dbfile","","");

sub foo {
   my $sth = $dbh->prepare( $some_query );
   $sth->execute;
}

sub bar {
    my $result = $dbh->do( $other_query );
}
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Or, for larger applications, see connect_cached() in the DBI documentation for a persistent DB connection handle.

Hope this helps!

update: showed alternative to connect_cached