discrete mountains, continuous functions?

One of the first inner geek moments recorded in the wild occurred in Montserrat Spain. If you are ever in the area and have a daytrip to spare, I highly recommend it. Not only is it a monastery in the mountains of features one of the few idols of a black virgin Mary as well as a fairly nice boys choir, they have a fairly fun set of hiking trails. Somewhere in the middle of things, we're probably deep into the part where we lost the trail and are bopping around in some random monk's digs. My mind starts going, and I remember a little trick from YSP back in Chicago.

The day starts with us reaching the base of the mountain at 10am. We hike up and down the mountain. There are switchbacks and sometimes we pick the wrong trail. It makes no difference what path we take, really. We ultimately get just short of the summit and setup camp for the night. We decide that we want to see the sunrise from the peak so we wake up at 6am the next day and hike to the top, where we relax, have breakfast, snap a picture, and make our way back down the mountain.

Rachel overlooking Montserrat

Continuing back down, we take a similar meandering path. Stop to pose for a picture, detour to escape a bear, we might have even left our camera at the top and had to have backtracked to retrieve it. After a grueling day of hiking, we stop at the gift-shop, buy a lousy t-shirt and then ride the tram back down into the city.

The geeky part of this that you can prove that with 100% certainty that there exists at least one time of day for which you were at the exact same altitude on both halves of the trip. More explicitly, there exists a time t in [00:00-23:59] such that f_0(t) = f_1(t) if f_i are functions of your altitude on the respective days of the hike.

Not convinced? Here are a few more clues:

We know that since we started our trip and ended our trip in the same place, so

f_0(00:00) = f_1(23:59)

We also know that we spent the night up in the mountain, so

f_0(23:59) = f_1(00:00)

The kicker is now that both functions f_0 and f_1 are continuous. There was no spot in time that we magically jumped altitudes, it was hectic, scattered even, but continuous nonetheless. Graphing these two functions on top of each other will show you that there is some value of t for which the two functions intersect. Bingo!

For further coolness, you can apply this same argument to the world and weather. One can easily accept that temperature when graphed out is also continuous. This allows me to say:

There exists at least one place (actually an uncountable number of places) for which the temperature at that location and the exact opposite, going through the center of the earth is the same.