Problem 71B#Problem 71B (Brouwer fixed point theorem). Use the previous problem to prove that any continuous function 𝑓:𝐷𝑛→𝐷𝑛 has a fixed point.Solution by kiwiyouAssume that 𝑓 has no fixed point. We can define 𝐹(𝑥) as the intersection of 𝑆𝑛−1 and the ray from 𝑓(𝑥) to 𝑥. The intersection is uniquely determined since 𝑥≠𝑓(𝑥).𝐷𝑛𝑆𝑛−1𝑥𝑓(𝑥)𝐹(𝑥)If such 𝑓 exists, the composition 𝑆𝑛−1↪︎𝐷𝑛→𝐹𝑆𝑛−1 is identity map, which is a contradiction.∎