A corollary of Mean Value Theorem

Question

This is from Mathematical Analysis by Browder.

The following is a corollary of Mean Value theorem

Let $U$ be a connected open set in $\mathbb{R}^n.$ If $f: U\to \mathbb{R}^m$ is differentiable at each point of $U,$ and $f'(p)=0$ for every $p\in U,$ then $f$ is constant.

Proof : Fix $p_0 \in U,$ and let $V = \{p\in U : f(p) =f(p_0)\}.$ Then $V$ is nonempty.

. . .and the proof goes.

How is $V$ non empty?

I know that a connected open set in $\mathbb{R}^n$ is path connected.

Answer 1

By assumption we have $p_{0}\in V$ that $f(p_{0})=f(p_{0})$, so $V\ne\emptyset$.