Let $f_1:\mathbb S^{n-1}\rightarrow \mathbb R^n$ be a continuous embedding, where $\mathbb S^{n-1}$ is the unit sphere of dimension $n-1$, and a point $x$ in the interior of $f_1(\mathbb S^{n-1})$ defined by Jordan-Brouwer separation theorem, let $$d:=\text{dist}(x,f_1(\mathbb S^{n-1})).$$ Now let $f_2:\mathbb S^{n-1}\rightarrow \mathbb R^n$ be another embedding such that $|f_1(y)-f_2(y)|<d$ for any $y\in \mathbb S^{n-1}$. Can we show that $x$ is still in the interior of $f_2(\mathbb S^{n-1})$? Another question: if we remove the assumption that $f_2$ is an embedding (just continuous), and we assume $f_2$ is the boundary of a continuous map $g: B^n\rightarrow \mathbb R^n$, can we show that $x\in g(B^n)$?