%EXERCISE TWELVE
          
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\begin{center}
	\textbf{Exercise Twelve: The }\verb!cases! \textbf{ and } 
	\verb!array! \textbf{ Environments}
\end{center}  

In a linear algebra course, we learn that the determinant of a 
$2 \times 2$ matrix is computed in the following way.

$$ 
\left | 
\begin{array}{cc} 
a & b \\
c & d 
\end{array}
\right |
=ad-bc
$$

Here's an interesting function. It's continuous \emph{only} at $x=0$.

$$ f(x)=
\begin{cases}
x, &\text{if } x\in\mathbb{Q}\\
-x, &\text{if }x\in\mathbb{R}-\mathbb{Q}
\end{cases}
$$

And here's a function whose mixed partials ($f_{xy}$ and $f_{yx}$) are not
equal at $(0,0)$.

$$ f(x,y)=
\begin{cases}
\frac{xy^3-yx^3}{x^2+y^2}, & \text{if } (x,y)\neq(0,0) \\
0, & \text{if } (x,y)=(0,0) \\
\end{cases}
$$

% The source of these 'pathological' counterexamples is
% http://www.math.tamu.edu/%7Etom.vogel/gallery/gallery.html

\end{document} 
