$5 \cdot 10$ The area of the closed region formed by the graph of the function $y=|||x|-1|-1|$ and the $x$-axis is
(A) 2.
(B) $\sqrt{2}$.
(C) 1.
(D) $\frac{1}{2}$.
(1st "Hope Cup" National Mathematics Invitational, 1991)
［Solution］Draw the graphs of
$$
\begin{array}{l}
y_{1}=|x|, \quad y_{2}=|x|-1, \\
y_{3}=|| x|-1|, \\
y_{4}=|| x|-1|-1 \\
y=|||x|-1|-1|
\end{array}
$$

It is easy to find that the area of the closed region formed by the graph of the given function and the $x$-axis is
$$
S=2 \cdot\left(\frac{1}{2} \cdot 2 \cdot 1\right)=2 \text {. }
$$

Therefore, the correct choice is $(A)$.