# Existence of certain continuous curves

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.