A person starts walking up a narrow mountain path at 6:00 a.m. The person walks at different speeds, stopping to eat, sometimes going back a few steps to look at a flower, but never leaving the path. The person eventually arrives at the top at 10:00 p.m. the same day, camps out over night and starts down the same path at 6:00 a.m. the next day. After stopping to eat and so forth the person arrives at the bottom at 10:00 p.m.

True or False: There is some point on the path that the person occupied at exactly the same time on the two different days. Explain your answer.