Continuity question on compact and connected domain

Let $\Omega$ be an open and bounded connected domain from $\mathbb{R}^N$. Consider a continuous function $f:\overline{\Omega}\to\mathbb{R}$. So $K=\overline{\Omega}$ is compact.

My question is: Can we find some $\delta>0$ such that $\forall\ x,y\in K$ with $|x-y|<\delta$ and $f(x)<0<f(y)$ there exists some $z\in K$ with $f(z)=0$ and $|x-z|\leq |x-y|$?

P.S. If $K$ is convex it is clearly true and easy to prove.