Dirac Delta Function: Contradictions

It is known that there exists the dirac delta function: $\delta(x) =\bigg\{\begin{array}{rr}\infty,\quad x = 0\\0,\quad x\neq 0\end{array}$

One of the properties of the delta function is "The distributional product of δ with x is equal to zero": $x\delta (x)=0$

As it is a property, it is assumed to be correct for every value of x.

However, when x = 0, $0\cdot \infty = 0$<--- What on Earth did I see???

Am I crazy???