Let $\mathbb{S}^n$ be the $n$-sphere. I would like to know if anyone knows of the following result in the literature (or whether anyone knows a proof/counterexample).

Let $f\colon\mathbb{S}^1\times[0,1]\to\mathbb{S}^2$ be a $C^1$-function such that $f(x,0)\ne p$ for all $x\in\mathbb{S}^1$. Then there exists a continuous function $g\colon[0,1]\to\mathbb{S}^2$ such that $g(0)=p$ and $f(x,t)\ne g(t)$ for all $(x,t)$.

The existence of space filling curves necessitates $f$ being a $C^1$-function, but other than that, I have not succeeded to construct a counterexample or a proof strategy/idea.


Here's a counter-example in the smooth case: take $p$ as the North pole, $f(\cdot,0)$ the Equator, and as $t$ grows, $f(\cdot,t)$ is a parallel of higher and higher latitude, until $f(\cdot,1)=p$.

If $f$ starts from the South pole instead and goes northwards to the North pole, no matter what $g(0)$ is chosen to be, it is impossible.


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