Let $X$ be an affine variety. $\text{Aut}(X)$ is the automorphism group of $X$. $\text{SAut}(X)$ is a subgroup spanned by all unipotent(image of additive group of field $K$) subgruops.
$X$ is called flexible if $\text{SAut}(X)$ act transitively on $X_{reg}$.
So I need to prove that if for affine variety $X$ there exists $f\in \mathbb{K}[X]^* \backslash \mathbb{K}^*$(nonconstant inversible regular function) then $X$ is not flexible.
I only got that if $f$ is nonconstant inversible function, then $f$ is non-constant on $X_{reg}$.
Any help is welcome :)
If $p,q\in X_{reg}$, your hypothesis says that you can connect it via finitely many curves, all of which are images of $\mathbb{A}^1$. Since $\mathbb{A}^1$ has only constants as units, one easily sees that $f(p)=f(q)$. Thus, $f$ is constant on $X_{reg}$ and since this is a dense open set, $f$ must be constant.