My question is this:
If $x\in \Omega=\{x:x\in\mathbb{R}^n, ||x||=1\} $, and it is given that $f: \Omega\to \mathbb{R}^n$ is Lipschitz, is the quantity $g(x) = x^T f(x)$ (or, for that matter any inner product, $g(x)=\langle x, f(x)\rangle$) Lipschitz too?
Are there any standard results to provide conditions for this?