Losing hope; becoming discouraged or hopeless, or describing someone who shows these characteristics.
From 'despond' plus the present participle suffix '-ing', functioning both as a verb form and as an adjective describing a hopeless state or person.
In older literature, a 'desponding youth' or 'desponding lover' became a romantic archetype—Victorian readers found something beautiful about characters wallowing in hopelessness, which seems bizarre until you remember Goth subculture did the same thing.
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