In Goldstein's classical mechanics, he makes an interesting claim, that if there is a continuous vector field $F$ where $F(\vec x) - F(\vec y)$ is parallel to $\vec x - \vec y$, then $F$ must be a constant field.
We can attempt a proof by contradiction. If such a non-constant field exists, it's clear that we can first choose some point $\vec x$ and decompose our vector field, into a component $F_{\vec x}^1$ that always points towards (or away from) $\vec x$ and another vector field $F^2_{\vec x}$ that is constant and equal to $F(\vec x)$. We can then repeat the construction with some other point $\vec y$.
I've used most of the information from the problem hypothesis, except continuity, and I'm not so sure how continuity and the above paragraph will yield a contradiction.
Goldstein's claim is from his chapter on rigid body motion, in a discussion on angular velocity. The claim appears just before equation $5.1$ of the third edition.
The claim is false, e.g. $F(\vec{x}) = \vec{x}$ satisfies the condition but is clearly not constant.
This error is actually pointed out in the article "Uniqueness of the angular velocity of a rigid body: Correction of two faulty proofs" by Nivaldo A. Lemos, where a correct proof is given.