To regard with disfavor or disapproval; to refuse to accept or acknowledge; to show disapproval of.
From dis- + countenance (which means 'face' or 'support' from Old French contenance). Originally meant to change someone's expression or refuse to look favorably upon them.
The word 'countenance' comes from the same root as 'face'—so 'discountenance' literally means to make someone lose face or show them you don't support them by changing your expression!
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