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Post Closed as "Not suitable for this site" by Steven Landsburg, Andy Putman, Daniele Tampieri, Mikhail Katz, Max Horn
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GH from MO
GH from MO
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Let $F: S^2 \rightarrow \mathbb{R}^2$ be a continuous function. Does there exist a uniteunit vector $v \in R^2$$v \in \mathbb{R}^2$ and a continuous functionsfunction $f(x):S^2\rightarrow \mathbb{R}$ such that $f(x)>0$ on $S^2$ and $$F(x)\neq f(x)v \ \ \ \ \forall x\in S^2?$$

Let $F: S^2 \rightarrow \mathbb{R}^2$ be a continuous function. Does there exist a unite vector $v \in R^2$ and a continuous functions $f(x):S^2\rightarrow \mathbb{R}$ such that $f(x)>0$ on $S^2$ and $$F(x)\neq f(x)v \ \ \ \ \forall x\in S^2?$$

Let $F: S^2 \rightarrow \mathbb{R}^2$ be a continuous function. Does there exist a unit vector $v \in \mathbb{R}^2$ and a continuous function $f(x):S^2\rightarrow \mathbb{R}$ such that $f(x)>0$ on $S^2$ and $$F(x)\neq f(x)v \ \ \ \ \forall x\in S^2?$$

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MathLearner
MathLearner
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Uncountable Cantor's diagonal argument on $S^2$

Let $F: S^2 \rightarrow \mathbb{R}^2$ be a continuous function. Does there exist a unite vector $v \in R^2$ and a continuous functions $f(x):S^2\rightarrow \mathbb{R}$ such that $f(x)>0$ on $S^2$ and $$F(x)\neq f(x)v \ \ \ \ \forall x\in S^2?$$