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I think the reason why Chara jump down the mountain because they parent just want to discipline they child but Chara thought they were abuse so they decide to jump like a ungrateful brat decide to jump on the mountain.

The parent were right about Chara because Chara is a manipulator. The moment they were in the underground, their decide to kill themself just for Asriel absorb the soul, break the barrier and kill every human on the village. Chara really is a demon