$\nabla U=0 \implies U=\mathrm{constant}$ with $A$ connected and $U$ continuous?

Question

I'm looking for suggestions on how to prove that

Given a function $U: A \subset \mathbb{R}^n \to \mathbb{R}$ with $U \in C^1 $ (therefore continuous) and $A$ a connected set, $$\nabla U=\bar{0} \,\,\,\,\,\,\, \forall \bar{x} \in A\implies U=\mathrm{constant} \,\,\,\,\,\,\, \forall \bar{x} \in A$$

In particular, how is the condition of $A$ as a connected necessary?