Let $L_t(s):S^1 \rightarrow \mathbb{R}^2(t\in [0,1])$ be a Jordan curve, $O(t)$ be its interior and $H(t,s)=L_t(s)$. If $H$ is a homotopy from $L_0(s)$ to$L_1(s)$, does there exist a continuous function $f:[0,1]\rightarrow \mathbb{R}^2$ s.t. $\forall t \in[0,1],f(t)\in O(t)$?