Show the area of the boundary of this set is not 0 In one real analysis book, it is said that: 

for the points set
  $S=\{(x,y)|0\le x\le 1, 0\le y\le D(x)\}$, where $D(x)$ is the Dirichlet function, $D(x)=
\begin{cases}
1,  & \text{if $x$ is rational} \\
0, & \text{if $x$ is irrational}  \\
\end{cases}$
  you cannot find the area of this set since the area of the boundary of this set, $\partial S=[0,1]\times[0,1]$ is not zero.

My question is how to show the area of boundary is not zero. 
Thanks!

What is the closure of $S$? 
Hint: If $x,y \in [0,1]\times [0,1]$ and $x_n \in [0,1] \cap \mathbb Q$ is such that $x_n \to x$ the you can show that $(x_n,y)\in S$.
What is the interior of $S$? Can you show that if $B$ is some ball inside $[0,1] \times [0,1]$ you can find some $(x,y)\in B$ so that $x \in \mathbb Q$?
These two together tell you what $\partial S$ is...
