The following statement is excerpted from the proof of a theorem about integral operator.
Let $D\subset{\mathbb R}^m$ be a bounded domain. If $\partial D$ is of class $C^1$, then the normal vector $\nu$ is continuous on $\partial D$. Therefore, we can choose $R\in(0,1]$ such that $ \nu(x)\cdot\nu(y)\geq\frac{1}{2} $ for all $x,y\in\partial D$ with $|x-y|\leq R$.
I can't understand the "therefore" part. Maybe the key point is in the connection between the continuity of $\nu$ and the "$\geq$" inequality. But I don't see any relationship between them. Can any one tell me what's the reasoning there?