1. The problem statement, all variables and given/known data 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) = f(c+1/2). Conclude there are, at any time, antipodal points on the earth's equator that have the same temperature. 2. Relevant equations 3. The attempt at a solution I need help working through this problem. I am confused on how to start. First off the problem feels like it completely changes tempo from discussing real analysis to jumping to earth's axis. What exactly are antipodal points? Are those the same as mid-points of subpoints in an interval? I assume so. Can someone tell me the first line in this proof after the assumptions so that i can get started?