way to include own functions independent of theme

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This is not quite simple (as copying and pasting into new functions.php :D), but you can create a plugin that will load this every time.

A (very) good and simple guide can be found here.

Addition by Original Poster (johannes)

I created a plugin for that:

  1. I created a simple php file and put it into wp-content/plugins. It looks basically like this (i.e. it contains the plugin name, description, author etc. and the function I want to use in my site across all themes):

    <?php
/**
 * Plugin Name: Function-Includer
 * Plugin URI: my URI
 * Description: my description
 * Version: 1.0
 * Author: johannes
 * Author URI: my URI
 */

[... code of my function ...]

?>

  2. Then I went into the WP admin backend, to the plugin page, where that plugin is listed as “Function-Includer” and activated it.

  3. Done – the function is available everywhere in my site now.