while studying signals & systems I ended up in the following difficult situation after digging up the unit step and delta functions a lot (I wish I hadn't do so, but couldn't help :]). Here it is:
If we define the unit step function as $$ u(t) = \left\{ \begin{array}{ll} 1 & \mbox{if } t \geq 0 \\ 0 & \mbox{if } t < 0 \end{array} \right. $$ the following holds I presume, $u(t)^2 = u(t)$.
Then, if we dare to take the derivative of both sides, it should be $$2u(t)\delta(t)=\delta(t)$$ which is also equivalent to $2u(0)\delta(t) = 2\delta(t)$. So, the RHS and LHS are not equal. But, they were.
So, since we're dealing with generalised functions/derivatives (which I'm not very familiar with) here, I'm making some serious mistakes in the operations above, but don't really know which ones.