Calling a Function in Functions.php from an external PHP script

You’re much better off with using a (mu-)plugin. Just add a folder named mu-plugins in your wp-content directory and place your files there. Then add a plugin header comment to it:

<?php
/**
 * Plugin Name: Some Name
 * Plugin URI: http://example.com
 * Description: Loads something
 * Version: 0.1
 * Author: Akamaozu
 * Author URI: http://example.com
 * License: GNU GPL 2 <https://gist.github.com/1365159>
 */
// deny direct file access if WP not loaded
! defined( 'ABSPATH' ) AND exit;

// define your functions here
function foo()
{
    print '<h1>'.__FILE__.' loaded successfully!</h1>';
}
add_action( 'shutdown', 'foo' );