I believe, and I may be wrong, $content_width in WordPress is a way to set the max width of images.
If you’re using responsive design techniques, what you could do is simply override the standard image width/height markup using CSS.
For example, setting a max-width of a particular amount, or indeed 100% – with height set to auto, that way you know that the images will never expand beyond their container, so when downsizing they will adjust to fit automagically.
Yes Andy, that’s exactly what I do with images. I too have read “rumours” of it being used to stop images overflowing, but it is so poorly documented. I thought I could just leave it out so I ran the theme-tester plugin after removing the:
if ( !isset( $content_width ) ) $content_width = 610;
and it threw up an error stating that the line was required so I don’t know what to do now.