= Solution
Use the <two-chain noncontinuous complete partial order>
$$
\boxed{P=\{\bot,\top\}\cup\{a_n:n\geq1\}\cup\{b_n:n\geq1\},}
$$
where $\bot<a_1<a_2<\cdots<\top$ and $\bot<b_1<b_2<\cdots<\top$, with no comparisons between the two interiors. In this solution $\sup D$ denotes the <least upper bound> printed as $\sqcap D$ in the question.<Calculus, linear algebra, discrete mathematics, logic and proof, and models of computation examination paper by the University of Oxford (2005)>[Printed p. 4, Q6, definitions preceding (a)]{ref}
Every nonempty <directed set> containing $\top$ has <least upper bound> $\top$. A <directed set> omitting $\top$ cannot contain both an $a_i$ and a $b_j$, because their only common <upper bound> in $P$ is $\top$, which would have to belong to that <directed set>. It therefore lies in one <chain>, possibly with $\bot$. If its indices are bounded it has a greatest element, which is its <least upper bound>; if they are unbounded its <least upper bound> is $\top$. Thus $P$ is a <directed-complete partial order>.
No element other than $\bot$ is <way-below> $\top$. For $a_i$, the <directed set> $\{b_n:n\geq1\}$ has <least upper bound> $\top$, but none of its members is above $a_i$. Interchanging the two <chains> excludes every $b_i$. Either <chain> excludes $\top$ itself. On the other hand, $\bot\ll\top$: every <directed set> whose <least upper bound> is above $\top$ is nonempty, and any member is above $\bot$. Hence
$$
\{y:y\ll\top\}=\{\bot\},\qquad\sup\{y:y\ll\top\}=\bot\ne\top.
$$
So $P$ is \b[not a <continuous directed-complete partial order>].
There is a small convention issue in the printed definitions. Our usual <directed set> is nonempty, but the question's pairwise condition also allows the empty <set>. Taken literally, the empty <set> has <least upper bound> $\bot$, and the preceding example and failure of continuity still work. Under this literal convention $\bot\not\ll\bot$, because the empty <set> supplies no witness. We retain this qualification in the next parts.
\Image[/past-exam-of-the-year-1-of-the-computer-science-course-of-the-university-of-oxford/2005/math-prelims-2005-order-examples.png]
{title=Two branches with a common supremum, compared with continuous approximation in the unit interval}
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