Intuitively, a map is determined on a region by its values on the points of a region. In the language of affine schemes, this suggests that $f(\mathfrak p_x)$ is determined by $\{f(\mathfrak m)\colon \mathfrak p_x\subset \mathfrak m\}$ (where we write $f(\mathfrak q):= f\bmod\mathfrak q$). But this isn't true, as can be seen by considering a local ring. Why is my intuition wrong?
($A$ is a commutative ring, $f\in A$, $x=\mathfrak p_x\in\operatorname{Spec}(A)$, $\mathfrak m$ is a maximal ideal of $A$)