At 6 am you start hiking on a path up a mountain. You walk at a variable pace, resting occasionally. At 6 pm that day you reach the top of the mountain. You camp out overnight.
The next morning you wake up at 6 am and start your descent down the mountain. Again you walk down the path at a variable pace, resting occasionally. The downhill hike is easier, but you are tired after yesterday and at 6 pm you reach the bottom.
Is it possible to find some time during the second day, such that you are at the exact same spot you were on at the same time on the first day?
You can always find a spot such that you were there at the same time on both the first and the second day.
Proof
Imagine a second hiker, who starts climbing the same path at 6 am on the second day. Now assume that the second hiker walks at exactly the same speed you did on the first, and takes the same breaks you did.
Now on the second day, since you both start at opposite ends of the path at 6 am and finish at opposite ends at 6 pm, you must have met each other along the way. The point where you meet gives a point where you were at the same position at a given time on both days.