Must There Be Two Years on a River Bank with the Same Temperature?

[chaoseverlasting]

Homework Statement

This is a question I had in a mathematics quiz I participated in today.

The river bank of a river on the west coast is under observation perfectly. Suppose the temperature variation for the last one hundred years is known absolutely. The temperature now is exactly as it was 2007 years ago. Must there be at least one pair of years, separated by one year, which have the exact same temperature?

Homework Equations

The Attempt at a Solution

I said yes as I thought the temperature change during a year would be a periodic function with period one (year). How could I prove or disprove it?

 

[EnumaElish]

The other interpretation is that "temperature" means "yearly average."

 

[chaoseverlasting]

If you can't make that assumption, how would you prove/disprove it? I was thinking of something along the rolle's or langrange's theorem assuming that the temperature is a continuous non monotonous function.

Even if its not periodic, the fact the temperature was the same 2007 years ago satisfies one condition of the rolle's theorem, therefore atleast one point must exist where f'(t)=0 (there should be many many points like that because of the temperature fluctuations), but how do you prove that there must be a pair of years with the same temperature one year apart?

 

[EnumaElish]

If temp. is a periodic function with constant yearly average, then it is kind of an absurd question. Either the question is missing information or it is relying on material that was covered in class that I am not privy to.

Otherwise, the answer may be either yes or no depending on the assumptions one makes.