Hello and welcome to just answer.
I am sorry to hear about your discomfort.
Black discoloration indicates that your tooth is definitely decayed.The main cause of tooth decay is presence of acid on the surface of teeth
Bacterial plaque is source of acid secretion which stimulates tooth decay and causes this bad smell.
Decay inside a tooth creates a situation where there is little oxygen and it creates a favorable envoirment for the sulfur metabolizing bacteria i.e the anaerobes to act on the decayed matter which gives rise to unpleasant smell and taste.
Trapped food particles between the teeth also all encourage the growth of growth of bacterial organisms and make the taste worse.
Loss of filling can also let seepage of bacterial products and saliva which can contribute to the progression of caries.
There is also possibility of secondary caries in your previously filled tooth thereby causing decay which appears black as you can see now when half of it is broken. These caries occur under the restoration and they continue to undermine the tooth surface.
So, you are correct and you have assumed it right...Unpleasant taste is because of the decayed tooth which needs to be evaluated and re-filled by your dentist as soon as possible.
See the link for some detailed information on this topic:
I hope this helps!
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The raised white dots are the normal anatomy of the tongue the papillae-fungiform and at the extreme back are the circumvallate papilla which are about to 8-10 in number.These contain taste buds and are not related to the bad taste you are having in your mouth,probably you are noticing them now.