Postgres 12

By using db<>fiddle, you agree to license everything you submit by Creative Commons CC0.
CREATE TABLE learningactivity (room_id text, weekday text, start_time int, end_time int);

INSERT INTO learningactivity  VALUES
  ('A', 'Monday', 09, 10)
, ('A', 'Monday', 10, 11)
, ('A', 'Monday', 13, 14)
, ('A', 'Monday', 14, 16)
;

CREATE OR REPLACE FUNCTION iweekday(weekday text)
  RETURNS int LANGUAGE sql IMMUTABLE PARALLEL SAFE AS
$func$
SELECT CASE weekday
         WHEN 'Monday'    THEN 1
         WHEN 'Tuesday'   THEN 2
         WHEN 'Wednesday' THEN 3
         WHEN 'Thursday'  THEN 4
         WHEN 'Friday'    THEN 5
         WHEN 'Saturday'  THEN 6  
         WHEN 'Sunday'    THEN 0
        END
$func$;

CREATE OR REPLACE FUNCTION fourpointnine()
  RETURNS TABLE (room_id text, iw int, start_time int, end_time int)
  LANGUAGE sql AS
$func$
   WITH RECURSIVE cte AS (
       SELECT l.room_id, l.weekday, l.start_time, l.end_time
       FROM   learningactivity l

       UNION ALL 
       SELECT l.room_id, l.weekday, c.start_time, l.end_time
       FROM   cte c
       JOIN   learningactivity l ON l.room_id = c.room_id
                                AND l.weekday = c.weekday
                                AND l.start_time = c.end_time
      )
   SELECT DISTINCT ON (1, 2)
          c.room_id, iweekday(c.weekday) AS iw, c.start_time, c.end_time
   FROM   cte c
   ORDER  BY 1, 2, c.start_time - c.end_time
$func$;
4 rows affected

WITH RECURSIVE cte AS (
   SELECT l.room_id, l.weekday, l.start_time, l.end_time
   FROM   learningactivity l

   UNION ALL 
   SELECT l.room_id, l.weekday, c.start_time, l.end_time
   FROM   cte c
   JOIN   learningactivity l ON l.room_id = c.room_id
                            AND l.weekday = c.weekday
                            AND l.start_time = c.end_time
  )
SELECT DISTINCT ON (1, 2)
       c.room_id, iweekday(c.weekday) AS iw, c.start_time, c.end_time
FROM   cte c
ORDER  BY 1, 2, c.start_time - c.end_time;


room_id
iw
start_time
end_time


A
1
13
16


SELECT * FROM fourpointnine();


room_id
iw
start_time
end_time


A
1
13
16