Question

Any continuous function on unit circle has at least a fixed point

Answer 1

Answer: Let x∈S1x∈S1 be not in the range of f:S1→S1f:S1→S1, and let us introduce a (constant-speed) parameterization ϕ:[0,1)→S1ϕ:[0,1)→S1 such that ϕ(0)=xϕ(0)=x. Then we consider F=ϕ−1∘f∘ϕ:[0,1)→[0,1)F=ϕ−1∘f∘ϕ:[0,1)→[0,1). We can continuously extend FF to F:[0,1]→[0,1]F:[0,1]→[0,1], and obviously, we have F(0)>0F(0)>0 and F(1)<1F(1)<1. Hence FF has a fixed point.

Step-by-step explanation:

Let f:S1→S1 be a continuous function, where S1 is the unit circle. Prove that if f is not onto, then f must have a fixed point.

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Answer 2

Answer:

Step-by-step explanation:

Let x∈S1 be not in the range of f:S1→S1, and let us introduce a (constant-speed) parameterization ϕ:[0,1)→S1 such that ϕ(0)=x. Then we consider F=ϕ−1∘f∘ϕ:[0,1)→[0,1). We can continuously extend F to F:[0,1]→[0,1], and obviously, we have F(0)>0 and F(1)<1. Hence F has a fixed point.