Obviously you wouldn't want to put that back on the front of the lens. But it led us to wonder what would happen if you did. So we tried it. In case you're interested, the lens was a Canon EF 85mm f/1.2 L II USM (we're glad she didn't drop that!!) and the camera we used was a Canon EOS 5D Mark III. All the images below are JPEGs straight out of the camera on its default settings, with no additional processing.
Firstly, with the lens stopped down to f/10, which is about the smallest aperture most people will be using most of the time. This is without the filter:
... and this is with the broken filter in place:
You might be surprised by how little difference there is. The second photo has noticeably reduced colour saturation and contrast, and that particularly affects the specular highlights on the metallic objects, but that's the main difference. The second photo also looks a little bit less sharp, but we reckon that's largely an effect of the reduced contrast. To show that, we can look at a couple of crops from the centres of the images. Firstly without the filter:
and then with the broken filter:
You can see that, despite the reduced contrast, there's still plenty of detail in the photo taken with the filter. (Remember, if you're viewing this on a standard desktop monitor, the full image would be about 4-5 feet across at this magnification!) We reckon that, if you were to boost the contrast on the with-filter photo, and sharpen them both judiciously, you'd really struggle to tell them apart.
And then we tried the same thing with the lens wide open at f/1.2. This is taken without the filter:
and this is taken with the broken filter in place:
Again, the main difference is a loss of contrast. The shallow depth of field pretty much masks all other effects. (Plus, the shallow depth of field makes the comparison a little bit more difficult: if you look closely at the tool kit you'll see that the second photo is focussed slightly further back than the first one. You wouldn't spot that at f/10.)
Wow. Remember, this is the filter which we used:
Some people are very sniffy about using protective filters because they degrade the image quality. Well, yes, they do. In some other applications - for example if you're shooting into the light, or shooting at night with lots of bright point sources of light - they can have a significant effect. But this little experiment demonstrates that, for most photography with "ordinary" subjects and "ordinary" lighting, even a totally smashed filter has surprisingly little effect. An undamaged filter would obviously have even less effect. And dust on the front of the lens ... don't worry about it!