The controller functions that you're trying to call, expect a parameter $slug. So the url to call the function should look like this:
PHP Code:
site_url() . 'karte/get_province/' . $slug;
In your code, the function never gets that parameter when you do an AJAX request.
To solve this, pass the slug as one of the post data fields.
Code:
data: {'tb_edition_id' : tb_edition_id, 'slug' : slug },
Of course, you'll have to let your javascript fetch the value of slug.
Then, in the controller, simply read the value with $this->input->post('slug');