Actually, few databases support a only time or only date datatype. It's better to NOT use Date or Time and just use datetime. It's also easier as you only need to update one column instead of two.
In MySQL, timestamp is updated to now() on any record update where that timestamp field is not set to a specific value, BUT, other databases, such as Postgresql, handle them differently. In pgsql, the timestamp is set on an insert but is not updated on an update automaticly.