Problem. Monotonicity is insufficient [dt-001K]

Liam O'Connor

Consider this monotonic function $g$ defined over a cpo $(\mathbb {R} \cup \{\infty ,-\infty \}, \leq )$ : $g(x) = \begin {cases} \tan ^{-1} x & \text {if}\ x < 0\\ 1 & \text {otherwise} \end {cases}$ Note this function is not continuous at 0. Starting from our $\bot$ element $-\infty$ , we can see that the limit of the ascending Kleene chain is $0$ : $\begin {array}{lcl} g(-\infty ) & = & -\frac {\pi }{2}\\ g(-\frac {\pi }{2}) & = & -1\\ g(-1) & \approx & -0.78\\ \end {array}$ But $g(0) = 1$ ! — the lub of the Kleene chain is not a fixed point!

Problem. Monotonicity is insufficient [dt-001K]

Problem. Monotonicity is insufficient [dt-001K]

Definition. Continuity [dt-001J]

Theorem. Continuity implies monotonicity [dt-001L]

Proof.

Example. Monotonic but not continuous [dt-001M]

Exercise. [dt-001N]

Solution.

Theorem. Alternative characterisation of continuity [dt-001O]

Proof.

Thesis.

Exercise. [dt-001W]

Solution.

Theorem. The Kleene fixed point theorem [dt-001P]

Proof.

Definition. The $\textbf {fix}$ operator [dt-001Q]

Exercise. [dt-001S]

Solution.

Solution.

Solution.