"Dirac Comb" vs "Ones Comb"
the integral

$$ \int\limits_{-a}^{a} \delta(x) \, \mathrm{d}x = 1  \qquad \forall a&gt;0$$

but this integral

$$ \int\limits_{-a}^{a} g(x) \, \mathrm{d}x = 0  \qquad \forall a&gt;0$$

where 

$$ g(x) = \begin{cases}
1 \qquad &amp; \text{for } x=0 \\
0 \qquad &amp; \text{for } x\ne0 \\
\end{cases} $$

this is why we need this dirac delta thing for the sampling or sifting property:

the integral

$$ \int\limits_{-\infty}^{\infty} f(x) \, \delta(x-x_0) \, \mathrm{d}x = f(x_0) $$

but this integral

$$ \int\limits_{-\infty}^{\infty} f(x) \, g(x-x_0) \, \mathrm{d}x = 0 $$

can you see what's wrong with using $g(x)$ for sampling?