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Can we classify all uniformly continuous functions from $X=S^1$-$\{(0,1), (0,-1)\}$ (circle -two points) to $1$-dim Euclidean space $\mathbb R$.

I feel that functions which take both the arcs of the circle to a fixed interval (same interval ) of $\mathbb R$ are uniformly continuous and these are the only uniformly continuous functions.

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