To continue in a place or situation, while others leave.(verb)
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Use "stay on" in a sentence
"LBJs efforts to ensure that Bundy would stay on included inviting Bundys mother for a one-on-one conversation in the Oval Office when he learned that she was in town visiting her son."
"Kennedy asked Eisenhower to postpone Goodpasters follow-on military assignment so that he could stay on for the new administrations first month; but he had no intention of keeping the elaborate NSC machinery in place."
"Instead, we did our best to stay on the safer side of Beulah's barrel."