When you divide [0,1] into sub-intervals, there will only be one sub-interval containing 1/2. The upper sum will be the length of this interval, the lower sum will be zero. The lower integral is obviously zero. The challenge is to prove that the upper integral is zero.
It shouldn't be hard to find a partition of [0,1] for which the upper integral is less than for any . You can use this to prove that the upper integral is zero.