First of all, we need to understand that the function of the temperature is continuous. In other words, the temperature changes gradually, either increasing or decreasing. It’s impossible for a 100 degree Celsius object become 0 degree Celsius at the next moment. The energy must be absorbed or released continuously, or a little by a little. Thus, all values on the graph of the temperature are connected without any jumps.
Since the temperature is changing gradually from two random points, position a to position b, we are able to find two places x in each side of the interval a and b and its temperature is between T(a) and T(b). The distance between a-x and b-x must be equal, because the temperature changes at the same rate. This means that these two places x are opposite to each other. The line of two x also passes through the center of the circle. Therefore, the temperatures of two antipodal points are the same.