Local maxima/minima and removable discontinuities

I think there is a minimum. It's just not at 0. Just because one point is removed from the function doesn't mean that the next best point(s) aren't minima. If you search on the interval (0,∞, you'll find that the minimum is very close to 0, but if you search on the interval [0,∞, you'll hit the problematic hole.

Not by a strict definition of minimum: f has a (global) minimum at x* if f(x*) ≤ f(x) for all x (in the domain of f).

For any x* > 0, f(x*) = x*^2. But then f(x*/2) = x*^2/4 = f(x*)/4 < f(x*), so x* is not a minimum.