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The character Tithonus, of the Greek legend Eos and Tithonus, managed to get eternal life. However, in the process, he forgot to ask for eternal youth along with it. So while he continued to age, he just would never actually die. Eventually Eos had to lock him in a room, due to frailty and senility.
The term Tithonus Syndrome refers to the fact that just about any source where immortality is presented, such as science fiction, fantasy, or mythology, presents it as a negative thing. That when a character is given immortality, there are always negative consequences, suffering, and the like.
Examples include the myth of the vampire, who needs to feast on blood. This theme has been often imitated in various fiction, where the antagonist(s) would hurt/kill others for some important chemical/element that would continue to keep them alive. There was also the woman, Naria, in Metropolis, who was transformed into a robot for her immortality - once again, a sacrifice of part of who the person is to achieve immortality.
Another common negative portrayal involves accelerated aging. Usually when the secret potion or elixir runs out, the person ages rapidly, to either be what they should be had they naturally aged to the same chronological age, or aging to death in a short amount of time.
The other most common portrayal is the one it's named after, where a person has immortality or some extreme length of life, but ages as normal, soon being unable to really do anything except lie there in suffering. | 
