Scott Continuous Map

A function $f:D\to E$ between Directed-Complete Partial Order is Scott continuous if it preserves all Joins of Directed Sets.

Properties

• Every Scott Continuous Map is Monotone.

  Suppose that $x\leq y$, and note that $y$ is the Join of the directed subset $\left\{\;x,y{\mid\;}\right\}$. This means that $f(y)$ is also the Join of the subset $\left\{\;f(x),f(y){\mid\;}\right\}$, so $f(x)\leq f(y)$.