Why $0 \delta(0)=0$? I defined the "function":
$$f(t)=t \delta(t)$$
I know that Dirac "function" is undefined at $t=0$ (see http://web.mit.edu/2.14/www/Handouts/Convolution.pdf).
In Wolfram I get $0 \delta(0)=0$ (http://www.wolframalpha.com/input/?i=0*DiracDelta(0)). Why? I expect $0 \delta(0)=undefined$ (if $\delta(0)=\infty$, thus I will have an indeterminate form $0 \infty$).
Thank you for your time.

Look at $\delta$ as distribution: $\langle \delta, f \rangle = f(0)$. Then $\langle t \delta(t), f(t) \rangle = \langle \delta (t), t f(t) \rangle = (tf(t))\mid_{t = 0} = 0 \cdot f(0) = 0$ (see multiplication of distribution by smooth functions).
