[sql] Using BETWEEN in CASE SQL statement

I want to get the avarage rate for all 12 months from our rate table and divide it by months, i started writing an SQL select with case, but i seem to be doing something wrong in the "Between" part..here's my SQL

SELECT AVG(SELL_RATE),
       AVG(BUY_RATE),
       CASE MONTHS
            WHEN RATE_DATE( BETWEEN '2010-01-01' AND '2010-01-31') THEN 'JANUARY'
            ELSE 'NOTHING'
   END AS 'MONTHS'
FROM   RATE
WHERE  CURRENCY_ID = CURRENCY -033'

You do not specify why you think it is wrong but I can se two dangers:

BETWEEN can be implemented differently in different databases sometimes it is including the border values and sometimes excluding, resulting in that 1 and 31 of january would end up NOTHING. You should test how you database does this.

Also, if RATE_DATE contains hours also 2010-01-31 might be translated to 2010-01-31 00:00 which also would exclude any row with an hour other that 00:00.