derivate of a piecewise function $f(x)$ at$ x=0$. There is a piecewise function $f(x)$
$$f(x)= \begin{cases} 1 ,\ \ \text{if}\ \  x \geq \ 0 \\ 0,\ \  \text{if}\ \  x<0 \end{cases}$$ 
what is the derivative of the $f(x)$ at $x=0$?
Is it $0$? Or since it is not continuous, the derivative does not exists? 

This is the Dirac's delta function.
http://en.wikipedia.org/wiki/Dirac_delta_function
http://mathworld.wolfram.com/DeltaFunction.html
So the answer to your question is that at the origin the derivative is unlimited.
$\lim\limits_{x \rightarrow 0}{\delta(x)}=\infty$
