From what I can tell, the goldfish seems to be housed in a very small container. Common goldfish are supposed to grow to well over 12" and should reach up to 24". I can't tell if it's a comet or not... but if it is, it won't grow as large as a common but it will still grow to over 12". If not kept in a proper sized home while developing, it will become stunted and suffer from stress due to the stunting and the ecology that caused the stunting. When fish get stressed, their immune systems falter and they succumb to various pathogens that they would normally have an immunity or resistance to. Here's my blog - Goldfish Care Sheet - http/goldlenny.blogspot.com/2007/03/gold...y-goldfish.html
It's hard to tell what the black is on those pictures so it could simply be a color change but it could also be Melanophores ( http/links.jstor.org/sici?sici=0006-3185...%3E2.0.CO%3B2-E ) which is the resulting black pigmentation that shows up around an injury that is healing. The injury could be a simple scrape, sores left from bites of a parasite or a lesion caused by a bacterial issue.
Has the fish had any kind of sickness or illness in that area?
Check the ammonia levels as it looks like the after effect of an ammonia burn. The area turns black as it starts to heal, and yes the fish needs at least 20 gallons