In low spirits from loss of hope or courage; extremely dejected and discouraged. It describes a state of deep discouragement where one feels there is little reason for optimism.
From Latin 'despondere' meaning 'to give up, lose heart' (de- 'away' + spondere 'to promise'). The original sense was of giving up one's promises or commitments due to hopelessness.
Despondent contains 'despond' - when you despond, you give up hope! Think of someone who's 'sponded' (responded) to life with 'de-' (down) - they're down in their response to everything.
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