Is the function, $f(z)=iz\bar{z}$ analytic?

If the Cauchy-Riemann equations were satisfied in an open set $D$, you could conclude that your function was analytic in $D$. But here the C-R equations are only satisfied at one point. That does not say your function is analytic at that point, it says your function is not analytic at all, because to be analytic at a point (according to the definition) it must be analytic in a neighbourhood of the point.