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Built into Drupal core, in /includes/theme.inc the function theme_breadcrumb, automatically inserts » or » into your breadcrumbs.

The function looks like this:

function theme_breadcrumb($variables) {
  $breadcrumb = $variables['breadcrumb'];

  if (!empty($breadcrumb)) {
    // Provide a navigational heading to give context for breadcrumb links to
    // screen-reader users. Make the heading invisible with .element-invisible.
    $output = '<h2 class="element-invisible">' . t('You are here') . '</h2>';

    $output .= '<div class="breadcrumb">' . implode(' » ', $breadcrumb) . '</div>';
    return $output;
  }
}

So if you want to change or remove the separator, all you have to do is redo this function in your template.php file within your theme. Make sure you rename the function THEME_breadcrumb, where THEME is replaced by the machine name of your theme. Here is an example where I replace the » with a space.

function THEME_breadcrumb($variables) {
  $breadcrumb = $variables['breadcrumb'];

  if (!empty($breadcrumb)) {
    // Provide a navigational heading to give context for breadcrumb links to
    // screen-reader users. Make the heading invisible with .element-invisible.
    $output = '<h2 class="element-invisible">' . t('You are here') . '</h2>';

    $output .= '<div class="breadcrumb">' . implode(' ', $breadcrumb) . '</div>';
    return $output;
  }
}

Now it will use this function instead of the core function, all done without hacking core.