Say the length of the path is L and the person starts at time 0 and ends at time T each day.

On the first day, let the distance from the base of the mountain to his location at time t be f(t), so f(0) = 0 and f(T) = L.

On the second day, let the distance from the base of the mountain to his location at time t be g(t), so g(0) = L and g(T) = 0.

f and g are continuous. What can you say about the function f-g?