Assume $S\colon B_1(0)\rightarrow \mathbb R^n$, where $B_1(0)$ is the unit ball of $\mathbb R^n$, and $$ S(x)=Ax $$ where $A$ is a given $n\times n$ matrix. Besides, continue maps $S_\varepsilon\colon B_1(0)\rightarrow \mathbb R^n$ uniformly converge to $S$ on $B_1(0)$ as $\varepsilon \rightarrow 0$. What is the weakest condition make the $S_\varepsilon$ has zero point for small enough $\varepsilon$ ?
In fact, when $\lvert A\rvert \ne0$, I feel, for small enough $\varepsilon$, $S_\varepsilon$ will has zero point. But I think this condition is too strong, seemly, just $a_{11}\ne0$ is enough, but I don't know how to show it.
After some test, only $a_{11}\ne0$ is not enough.