package My::DBI;
    use strict;

    sub new {
        my $class = shift;
        bless { @_ }, $class;
    }

    sub connect {
        my $self = shift;
        return $self->{dbh} if defined $self->{dbh};
        croak "Need connection parameters"
            unless defined $self->{connect};

        my $c = $self->{connect};   ## Simplify typing :)
        return $self->{dbh} = DBI->connect
            ($c->{dsn}, $c->{user}, $c->{pass}, 'mysql',
             { RaiseError => 1 });
    }

    sub my_function {
        my $self = shift;
        ## $DBH is now in $self->{dbh}
    }

    ...

##</code><code>##

    my $db = My::DBI->new(connect => { ... });
    $db->my_function(@args);