I need a script that will take a pdb file with discontinuities and will give a pdb file with the same format but with a continuous res number

I'm assuming here that every time the residue name changes, the residue number is incremented. And in fact, that later on, the same residue name may occur again (with some intervening lines with different residue names), but should have a different number.

I would rework your original script to something like this.

#!/usr/bin/env perl

# Read a PDB file and change the residue numbers to be continuous.

use strict;
use warnings;

print "\nEnter the PDB input file: ";
my $inputFile = <STDIN>;
chomp $inputFile;

unless (open(INPUTFILE, "<", $inputFile)) {
    die "Cannot read from <$inputFile>, $!";
}

my $output_file = "my2pdb.txt";
open(PDBOUT,">>my2pdb.txt")
    or die "Cannot open <$output_file> for writing, $!";

my $last_residue_name = '';
my $last_residue_number = 0;

while (<INPUTFILE>) {
    if (m/^ATOM/) {
        my @m = split;

        # Increment the residue number if the residue name changes
        if ($m[3] eq $last_residue_name) {
            $m[5] = $last_residue_number;
        } else {
            $m[5] = ++$last_residue_number;
            $last_residue_name = $m[3];
        }
        printf PDBOUT "%4s %5s %4s %3s %1s %4s     %7s %7s %7s %6s %6s
+\n", @m;
    }
}

exit;
[download]

Here's the input file I used:

ATOM    310 1HH2 ARG A 607     -31.278  29.882  25.723  1.00 99.99
ATOM    311 2HH2 ARG A 607     -32.344  30.932  24.851  1.00 99.99
ATOM    312  N   LEU A 608     -36.327  31.914  18.187  1.00 65.62
ATOM    313  CA  LEU A 608     -37.435  32.634  17.559  1.00 67.47
ATOM    314  C   LEU A 608     -38.434  33.052  18.624  1.00 74.29
ATOM    315  O   LEU A 608     -38.982  32.201  19.331  1.00 71.12
ATOM    316  CB  LEU A 608     -38.110  31.803  16.459  1.00 64.64
ATOM    317  CG  LEU A 608     -39.261  32.481  15.719  1.00 71.07
ATOM    318  CD1 LEU A 608     -38.782  33.704  14.929  1.00 73.68
ATOM    319  CD2 LEU A 608     -39.981  31.498  14.829  1.00 69.63
ATOM    320  H   LEU A 608     -36.638  31.041  18.563  1.00 99.99
ATOM    321  N   ARG A 565     -38.634  34.587  18.911  1.00 22.27
ATOM    322  CA  ARG A 565     -39.655  35.200  19.766  1.00 23.04
ATOM    323  C   ARG A 565     -40.963  35.104  19.007  1.00 22.72
ATOM    324  O   ARG A 565     -41.046  35.500  17.847  1.00 24.21
ATOM    325  CB  ARG A 565     -39.275  36.643  20.105  1.00 99.99
ATOM    326  CG  ARG A 565     -38.044  36.770  20.986  1.00 99.99
[download]

And here's the output I got (note that column 5 is included, unlike your initial example):

ATOM   310 1HH2 ARG A    1     -31.278  29.882  25.723   1.00  99.99
ATOM   311 2HH2 ARG A    1     -32.344  30.932  24.851   1.00  99.99
ATOM   312    N LEU A    2     -36.327  31.914  18.187   1.00  65.62
ATOM   313   CA LEU A    2     -37.435  32.634  17.559   1.00  67.47
ATOM   314    C LEU A    2     -38.434  33.052  18.624   1.00  74.29
ATOM   315    O LEU A    2     -38.982  32.201  19.331   1.00  71.12
ATOM   316   CB LEU A    2     -38.110  31.803  16.459   1.00  64.64
ATOM   317   CG LEU A    2     -39.261  32.481  15.719   1.00  71.07
ATOM   318  CD1 LEU A    2     -38.782  33.704  14.929   1.00  73.68
ATOM   319  CD2 LEU A    2     -39.981  31.498  14.829   1.00  69.63
ATOM   320    H LEU A    2     -36.638  31.041  18.563   1.00  99.99
ATOM   321    N ARG A    3     -38.634  34.587  18.911   1.00  22.27
ATOM   322   CA ARG A    3     -39.655  35.200  19.766   1.00  23.04
ATOM   323    C ARG A    3     -40.963  35.104  19.007   1.00  22.72
ATOM   324    O ARG A    3     -41.046  35.500  17.847   1.00  24.21
ATOM   325   CB ARG A    3     -39.275  36.643  20.105   1.00  99.99
ATOM   326   CG ARG A    3     -38.044  36.770  20.986   1.00  99.99
[download]

(I don't do formats, you'll have to figure that out for yourself.)

Now that still begs the question of whether you wanted the entire input file written to the output, or just the ATOM section.

I'm assuming I've missed several hints about the final output, but this is the basic structure.