Yes. See what happens if you try to (fairly gently) bend the toe up.Looks like a fracture to me. A sprain wouldn't produce that level of brusing and swelling. At least, that's what my doctor said two months ago when I busted my ankle and it swelled up and went purple.
If broken = plenty of pain when the bone edges rub against each other.
It helps to tape the toe to the next one to immobilize it.