Well, the exact powers of a child in mythology would vary, but likely it would be centered around Agriculture, quite possibly going into the Harvest and The Seasons.
Demeter is a Goddess, an Olympian, a daughter of Kronos and Rhea. She is by no means a demigod.
She had an affair with the demigod Percy
When a god marries a demigod the demigod receives the same powers as the god. Which makes the demigod now more powerful the god. So the demigod has their powers and the powers of the god.
The Power Of Zeus Children is the power to throw lighting bolts.
The Greek goddess Artemis was a virgin goddess, so she had no children. If she did have a demigod child, her/his powers would be similar to Artemis'. They would probable be able to shoot an arrow with perfect aim and heal.
A demigod child of Apollo typically has powers related to music, archery, healing, prophecy, and some control over light. They may also possess heightened physical abilities, such as agility and stamina, and have a natural affinity for the arts.
A demigod child of Ares would have skilled control of weapons (without training), superiour horsemanship, and in many cases are master strategists.
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Demeter was born from the Titan gods Rhea and Cronus, she was born a Greek goddess, and so always had her "powers".
She is an Goddess. Of course she has powers.
She had none, she was a virgin goddess.
No, Perseus was a mortal. actually Perseus was a demigod (half god and half mortal) his father is Zeus and mother is Danae(a mortal)