Example of $f:\mathbb{R}^n\to \mathbb{R}^m$ such that $Df(x)^T f(x)=0$, for every $x\in \mathbb{R}^n$.

Question

Is there any function $f:\mathbb{R}^n\to \mathbb{R}^m$ such that $Df(x)^T f(x)=0$, for every $x\in \mathbb{R}^n$? It would be enough if this holds in some open subset of $\mathbb{R}^n$. I should mention that I look for examples where $Df(x)$ is continuous and not identically zero.

By $Df(x)$, I denote the Jacobian matrix of $f$ at $x$, that is, the matrix whose $(i,j)$-th entry is $\partial f_i(x)/\partial x_j$, and $Df(x)^T\in\mathbb{R}^n\times \mathbb{R}^m$ denotes its transpose (adjoint).

Thanks!

Answer 1

Consider $f : \mathbb{R} \to \mathbb{R}^2$ given by $$f(t) = \begin{pmatrix} \cos t \\ \sin t \end{pmatrix}.$$