An example of function that is not constant but its derivative is 0

Question

Let $U \subset R^{n}$ be and open set. Let $f:U \to R^n$ be a differentiable function.

I wonder if there is such a function that is not constant but with $Df=0$ at every point. (The derivative is 0).

Answer 1

Consider $f$ defined on $\mathrm{U} = \mathbf{R}^2 \setminus\{(x,0) : x\in \mathbf{R}\}.$ Here $\mathrm{U}$ is the plane minus the $x$-axis. Define $f = (0,0)$ on the upper part of $\mathrm{U}$ and $f = (-1,0)$ on the lower part.

Answer 2

Let $U = (0,1)\cup (2,3)$ and $f: U \rightarrow \mathbb{R}$ and $f(x) = 0$ for $x \in (0,1)$ and $f(x) = 1$ for $x \in (2,3)$. For open and connected sets $U$, $Df \equiv 0$ implies that $f$ is constant (I can provide a proof if you want).