SQL Server 2017

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WITH yourTable AS (
    SELECT '2018-04-01' AS Date UNION ALL
    SELECT '2018-04-02' UNION ALL
    SELECT '2018-04-18' UNION ALL
    SELECT '2018-04-19' UNION ALL
    SELECT '2018-04-21' UNION ALL
    SELECT '2018-04-22' UNION ALL
    SELECT '2018-04-23' UNION ALL
    SELECT '2018-04-24' UNION ALL
    SELECT '2018-04-25' UNION ALL
    SELECT '2018-04-26' UNION ALL
    SELECT '2018-04-27' UNION ALL
    SELECT '2018-04-28' UNION ALL
    SELECT '2018-04-29' UNION ALL
    SELECT '2018-05-05' UNION ALL

SELECT '2018-05-06' UNION ALL

SELECT '2018-05-07'
 UNION ALL
    SELECT '2018-05-08' UNION ALL
    SELECT '2018-05-09'
 UNION ALL
    SELECT '2018-05-28' UNION ALL
    SELECT '2018-05-29' UNION ALL

SELECT '2018-05-30' UNION ALL
    SELECT '2018-05-31'
),
cte1 AS (
    SELECT t1.Date, ROW_NUMBER() OVER (ORDER BY t1.Date) rn1,
       ROW_NUMBER() OVER (ORDER BY t1.Date DESC) rn2
    FROM yourTable t1
    WHERE NOT EXISTS (SELECT TOP 1 t2.Date FROM yourTable t2 WHERE t2.Date = DATEADD(day, -1, t1.Date)) OR
          NOT EXISTS (SELECT TOP 1 t2.Date FROM yourTable t2 WHERE t2.Date = DATEADD(day, 1, t1.Date))
)

SELECT DATEADD(day, 1, MIN(Date)) AS start_date, DATEADD(day, -1, MAX(Date)) AS end_date
FROM cte1
WHERE rn1 > 1
GROUP BY rn1 / 2
HAVING MIN(Date) <> MAX(Date)

start_date	end_date
03/04/2018 00:00:00	17/04/2018 00:00:00
20/04/2018 00:00:00	20/04/2018 00:00:00
30/04/2018 00:00:00	04/05/2018 00:00:00
10/05/2018 00:00:00	27/05/2018 00:00:00