Actually, in complex analysis, it's common to say that the function 1/z maps 0 to infinity when infinity is considered as the point at infinity on the Riemann sphere. So, in that sense, you could say 1/0 = infinity (when it's done naively by calculus students, this is wrong because they don't have a mapping in mind). You could restrict this to get 1/x when x is a real variable. The trick here is that you have to identify negative infinity with positive infinity. This isn't to say that 0 has an inverse. It is just that it is now included in the domain of the function 1/x and the point at infinity is added to the range.
But, still, 0/0 wouldn't have a good interpretation because that would correspond to the function 0/x, which is zero everywhere. I guess you could send 0 to 0, so that the function is continuous. So, you could define 0/0 to be zero. But it would be very confusing and bad notation that wouldn't accomplish anything, since there's no need to describe the constant function equal to 0 by such a convoluted means. And again, you would need to be careful to point out that it's a mapping, rather than taking an inverse, but that's a moot point. Better not to discuss it at all than to cause all this confusion. So, yes, 0/0 is undefined.