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
$$F(x)\neq f(x)v \ \ \ \ \forall x\in S^2?$$



Uncountable Cantor's diagonal argument on $S^2$
Sure. Simply choose $f(x)=\|F(x)\|+1$, and pick an arbitrary unit vector $v\in\mathbb{R}^2$.


