I am reading a Text about Single Index Models where a theorem is given for the identification in case all covariates are continuous. The theorem states these four conditions:
$G$ is differentiable and not constant on the support of $X'\beta$.
The components of $X$ are continuously distributed random variables that have a joint probability density function.
The support of $X$ is not contained in any proper linear subspace of $\mathbb{R}^d$.
$\beta_1 = 1$.
I understand condition 1, 2 and 4 because they were explained in the text. However, I do not understand condition 3. What does this condition mean and why is it important for identification?
So far, I can only guess why it is needed. Could it be possible that this is needed in order to ensure that the support of $X$ is $\mathbb{R}^d$ as a whole?