Let f be continuous on the interval [0, 1] to R and such that f(0) = f(1).

Question:

Let f be continuous on the interval [0, 1] to R and such that f(0) = f(1). Prove that there exists a point c in [0, 1/2] such that f(c + 1/2). [Consider g(x) = f(x) - f(x + 1/2).] Conclude that there are, at any time, antipodal points on the earth's equator that have the same temperature.