Puzzle #39: Get Out of Town!

I graduated with my baccalaureate in June 2010. This was my last puzzle of the 2009-2010 academic year:

According to [convocation link deleted], graduation for my program is on June 4, 2010 starting at 2 P.M. The earlier graduation ceremony that day starts at 10 A.M. The two graduation ceremonies presumably do not overlap and will be of similar lengths, so let us assume that my graduation ceremony will be no longer than four hours. Assume that I leave town right after graduation, taking fifteen minutes to reach the municipal boundary. Is it possible that I will still be in town at a time when the date and time expressed as YYMMDD hh:mm can have all ten digits different? There is a quick answer to this problem, but any valid reasoning will be accepted.