a) Let f(x) be continuous on  between numbers a and b. Then for any y between f(a) and f(b), there exists c between the numbers a and b. In this question, it claims that Pa is equal to the temperature at Pa+π by the Intermediate Value Theorem. The Intermediate Value Theorem works in the question because f(0) is smaller than 0 and f(π) is bigger than 0, or vice versa. Therefore, if there is a number a in between 0 and π, it means that there is f(a) between f(0) and f(π). When the function is continuous and there is a number between two other bigger and smaller numbers, it is impossible to the y-value of the number is bigger than y-value of the bigger number, and smaller than y-value of the smaller number in circular shaped graph. Therefore, there exists a number a in between 0 and π such that f(a) equals 0 and the temperature at Pa is equal to the temperature at Pa+π

b) My argument relies on the concept of continuity.