WordNet records no synonym and no antonym for discountenance. It is the only member of each of its synsets, which is common for technical, taxonomic and single-sense vocabulary — there is no other English word that means the same thing. The broader and narrower terms below are the useful relations in that case.
Counts are distinct word forms across every sense. A synonym that appears under three senses is counted once here and shown three times below, under each sense it belongs to.
look with disfavor on
“The republic soon discountenanced its few friends”
show disapproval by discouraging
“any measure tending to fuse invalids into a class with special privileges should be discountenanced”
Synonyms, antonyms and the broader and narrower relations for discountenance come from Princeton University’s WordNet. Full sources and licences.