In these notes on algebraic groups (http://www.jmilne.org/math/CourseNotes/iAG200.pdf), assume $(X, \mathcal O_X)$ is an algebraic $k$-scheme for some field $k$, and $S$ is a subset of $X(k)$, where $X(k)$ is the set of $x \in X$ for which the residue field $\kappa(x)$ is equal to $k$. A subset $S \subseteq X(k)$ is said to be schematically dense in $X$ if for any open subset $U$ of $X$, the family of maps $$f \mapsto f(s): \mathcal O(U) \rightarrow \kappa(s) = k; s \in S \cap U$$ is injective. I don't understand what the definition is saying. Is it saying that if we fix any $s \in S \cap U$, then the association $f \mapsto f(s)$ defines an injective function $\mathcal O(U) \rightarrow k$? That seems absurd. Or is it saying that if we fix any $f \in \mathcal O(U)$, and define a function $S \cap U \rightarrow k$ by $s \mapsto f(s)$, then this mapping is injective. The latter makes more sense, but the former seems to be the literal interpretation of what was written.
I'm trying to make sense out of this with an example. Let $k = \mathbb{Q}$, and let $X$ be the affine $k$-scheme whose underlying space is the set of maximal ideals of $k[T]$. Then $X$ is in bijection with the monic irreducible polynomials of $k[T]$, and has the cofinite topology. Also, $X(k)$ is the set of monic linear polynomials $T - x : x \in \mathbb{Q}$, then we can identify $X(k) = \mathbb{Q}$. Now if $S \subseteq k$ is schematically dense in $X$, then according to the second possible definition I mentioned, for any $f \in k[T]$, the map $s \mapsto f(s)$ should be injective, since $f(s)$ is the image of $f$ in the residue field $\mathcal O_{X,s}/\mathfrak m_s = \mathbb{Q}[T]/(T-s) \simeq \mathbb{Q}$. This is also absurd, so the second possible definition I mentioned doesn't make sense either.
The definition is stating that the map $f \mapsto \{f(s)\}_{s \in S}$ is injective, meaning that a function is completely determined by its values on $S$.
This is just the scheme theoretic version of the statement that a standard continuous function in topology (e.g. $\mathbf{R} \to \mathbf{R}$) is determined by its values on a dense subset.