MySQL 8.0

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CREATE TABLE tablename (
  `id` INTEGER,
  `user_id` INTEGER,
  `employment_type` VARCHAR(9),
  `date` DATETIME
);

INSERT INTO tablename
  (`id`, `user_id`, `employment_type`, `date`)
VALUES
  ('1', '9', 'full-time', '2013-01-01'),
  ('2', '9', 'half-time', '2013-05-10'),
  ('3', '9', 'full-time', '2013-12-01'),
  ('4', '248', 'intern', '2015-01-01'),
  ('5', '248', 'full-time', '2018-10-10'),
  ('6', '58', 'half-time', '2020-10-10'),
  ('7', '248', NULL, '2021-01-01');
SELECT DISTINCT user_id
FROM (
  SELECT user_id,
         FIRST_VALUE(employment_type) OVER (PARTITION BY user_id ORDER BY date DESC) last_type
  FROM tablename
  WHERE date <= '2014-01-01'
) t
WHERE last_type = 'full-time'


user_id


9


SELECT t1.user_id
FROM tablename t1
WHERE t1.date <= '2014-01-01' AND t1.employment_type = 'full-time'
  AND NOT EXISTS (
    SELECT 1 
    FROM tablename t2 
    WHERE t2.user_id = t1.user_id AND t2.date BETWEEN t1.date AND '2014-01-01' 
      AND COALESCE(t2.employment_type, '') <> t1.employment_type
  )


user_id


9