Here is an example to detect broken Images
http://maisonbisson.com/blog/post/12150/detecting-broken-images-in-javascript/
but is it possible to detect an broken background-image?
For exmaple:
<div style="background-image:url('http://myimage.de/image.png');"></div>
Do you have control over the "404 not found" page on your server? In that case you could point it to a script that looks for .jpg/.png/.gif in the requested URL and log accordingly, then outputting a 404 page to the user.
I dont think you need to have jQuery or javascript to tell you whether a link is broken. Use Firebug in Firefox and it will sort out most of your problems:
https://addons.mozilla.org/en-us/firefox/addon/firebug/
Edit: Now that I know that it was for an auto fix, i quickly had a look at it and came up with this:
Add that to your page after it has loaded and it will scan all div elements for any broken css background images. There might be a better or quicker way to do this but this is the general idea.
Edit 2: I noticed when i tested the script that .css('background-image') returns a string with "url()" enclosing the image url. This resulted in the ajax call failing on all urls. I changed it and actioned ajax calls on only elements which has css backgrounds. The above code now works perfect! :D